"Fingerprint","Nb Retrieved Papers"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The converse is obvious.""","21"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The converse is obvious.""","19"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""This is clear.""","18"
"""it is well known that"" AND ""it is not yet known whether"" AND ""a central problem in""","13"
"""under the additional assumption that"" AND ""it is well known that"" AND ""it has long been known that""","12"
"""This is trivial."" AND ""This is straightforward."" AND ""This is simple.""","11"
"""it is well known that"" AND ""it has long been known that"" AND ""a central problem in""","10"
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"""was a milestone in"" AND ""recently, there has been much interest in the"" AND ""have raised the question of whether""","9"
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"""was a milestone in"" AND ""the main result was the"" AND ""it would be interesting to apply the techniques of""","8"
"""under the additional assumption that"" AND ""recently, there has been much interest in the"" AND ""have raised the question of whether""","8"
"""recently, there has been much interest in the"" AND ""it would be interesting to apply the techniques of"" AND ""every student is aware that""","8"
"""under the additional assumption that"" AND ""the main result was the"" AND ""have raised the question of whether""","8"
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"""was a milestone in"" AND ""under the additional assumption that"" AND ""it is well known that""","8"
"""was a milestone in"" AND ""the groundbreaking work of"" AND ""have raised the question of whether""","8"
"""the main result was the"" AND ""recently, there has been much interest in the"" AND ""every student is aware that""","8"
"""the main result was the"" AND ""it would be interesting to apply the techniques of"" AND ""every student is aware that""","8"
"""we wish to extend the results of"" AND ""the main result was the"" AND ""have raised the question of whether""","8"
"""we wish to extend the results of"" AND ""it has long been known that"" AND ""have raised the question of whether""","8"
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"""was a milestone in"" AND ""under the additional assumption that"" AND ""recently, there has been much interest in the""","8"
"""the main result was the"" AND ""in this setting, the ability to"" AND ""have raised the question of whether""","8"
"""the main result was the"" AND ""recently, there has been much interest in the"" AND ""it would be interesting to apply the techniques of""","8"
"""We show the contrapositive."" AND ""We follow"" AND ""This is obvious.""","8"
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"""was a milestone in"" AND ""it would be interesting to apply the techniques of"" AND ""every student is aware that""","8"
"""was a milestone in"" AND ""recently, there has been much interest in the"" AND ""every student is aware that""","8"
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"""we wish to extend the results of"" AND ""unfortunately, we cannot assume that"" AND ""in this setting, the ability to""","8"
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"""we wish to extend the results of"" AND ""this reduces the results of"" AND ""recently, there has been much interest in the""","8"
"""the groundbreaking work of"" AND ""it is well known that"" AND ""a central problem in""","8"
"""we wish to extend the results of"" AND ""was a major advance."" AND ""the groundbreaking work of""","8"
"""we wish to extend the results of"" AND ""was a milestone in"" AND ""it has long been known that""","8"
"""This is trivial."" AND ""This is simple."" AND ""This is clear.""","8"
"""the main result was the"" AND ""it would be interesting to apply the techniques of"" AND ""have raised the question of whether""","8"
"""we wish to extend the results of"" AND ""was a milestone in"" AND ""have raised the question of whether""","8"
"""We proceed by induction."" AND ""This is trivial."" AND ""Suppose the contrary.""","8"
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"""it would be interesting to apply the techniques of"" AND ""have raised the question of whether"" AND ""every student is aware that""","8"
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"""was a major advance."" AND ""the groundbreaking work of"" AND ""have raised the question of whether""","8"
"""we wish to extend the results of"" AND ""the main result was the"" AND ""it has long been known that""","8"
"""the main result was the"" AND ""it has long been known that"" AND ""have raised the question of whether""","8"
"""the main result was the"" AND ""the groundbreaking work of"" AND ""have raised the question of whether""","8"
"""we wish to extend the results of"" AND ""this reduces the results of"" AND ""the groundbreaking work of""","8"
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"""was a major advance."" AND ""the groundbreaking work of"" AND ""recently, there has been much interest in the""","8"
"""This is a contradiction."" AND ""The converse is trivial."" AND ""The converse is obvious.""","8"
"""was a milestone in"" AND ""in this setting, the ability to"" AND ""have raised the question of whether""","8"
"""it is well known that"" AND ""it has long been known that"" AND ""in this context, the results of""","7"
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"""under the additional assumption that"" AND ""recently, there has been much interest in the"" AND ""it has long been known that""","7"
"""This is straightforward."" AND ""This is simple."" AND ""This is obvious.""","7"
"""we wish to extend the results of"" AND ""was a milestone in"" AND ""under the additional assumption that""","7"
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"""this reduces the results of"" AND ""the main result was the"" AND ""it is essential to consider that""","5"
"""this reduces the results of"" AND ""this leaves open the question of"" AND ""it would be interesting to apply the techniques of""","5"
"""the groundbreaking work of"" AND ""recently, there has been much interest in the"" AND ""it is well known that""","5"
"""the main result was the"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","5"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""every student is aware that""","5"
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"""unfortunately, we cannot assume that"" AND ""recently, there has been much interest in the"" AND ""a central problem in""","5"
"""unfortunately, we cannot assume that"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","5"
"""unfortunately, we cannot assume that"" AND ""in this context, the results of"" AND ""a central problem in""","5"
"""was a major advance."" AND ""this could shed important light on a conjecture of"" AND ""the main result was the""","5"
"""this reduces the results of"" AND ""it is essential to consider that"" AND ""every student is aware that""","5"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of""","5"
"""who first asked whether"" AND ""was a major advance."" AND ""improved upon the results of""","5"
"""who first asked whether"" AND ""was a major advance."" AND ""the main result was the""","5"
"""who first asked whether"" AND ""it would be interesting to apply the techniques of"" AND ""in future work, we plan to address questions of""","5"
"""who first asked whether"" AND ""this reduces the results of"" AND ""it is essential to consider that""","5"
"""the groundbreaking work of"" AND ""it is not yet known whether"" AND ""every student is aware that""","5"
"""who first asked whether"" AND ""this could shed important light on a conjecture of"" AND ""in future work, we plan to address questions of""","5"
"""who first asked whether"" AND ""the groundbreaking work of"" AND ""improved upon the results of""","5"
"""in future work, we plan to address questions of"" AND ""improved upon the results of"" AND ""every student is aware that""","5"
"""unfortunately, we cannot assume that"" AND ""it is well known that"" AND ""every student is aware that""","5"
"""it is essential to consider that"" AND ""it has long been known that"" AND ""in future work, we plan to address questions of""","5"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""a central problem in""","5"
"""under the additional assumption that"" AND ""it would be interesting to apply the techniques of"" AND ""in this context, the results of""","5"
"""was a major advance."" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""under the additional assumption that""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""it is not yet known whether""","4"
"""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""was a major advance."" AND ""it would be interesting to apply the techniques of"" AND ""it is essential to consider that""","4"
"""was a milestone in"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""We proceed by induction."" AND ""This is straightforward."" AND ""Suppose the contrary.""","4"
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"""unfortunately, we cannot assume that"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","4"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""have raised the question of whether""","4"
"""the groundbreaking work of"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""the groundbreaking work of""","4"
"""this could shed important light on a conjecture of"" AND ""the groundbreaking work of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""it would be interesting to apply the techniques of""","4"
"""the groundbreaking work of"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""in this setting, the ability to""","4"
"""was a major advance."" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""unfortunately, we cannot assume that"" AND ""under the additional assumption that"" AND ""this leaves open the question of""","4"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""this could shed important light on a conjecture of"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""improved upon the results of""","4"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""in future work, we plan to address questions of""","4"
"""this could shed important light on a conjecture of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""this reduces the results of"" AND ""it is essential to consider that""","4"
"""under the additional assumption that"" AND ""this reduces the results of"" AND ""in this setting, the ability to""","4"
"""it is essential to consider that"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""it is well known that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""a central problem in""","4"
"""this could shed important light on a conjecture of"" AND ""it is not yet known whether"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""it would be interesting to apply the techniques of"" AND ""in this context, the results of""","4"
"""this reduces the results of"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""in future work, we plan to address questions of""","4"
"""we wish to extend the results of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""it has long been known that""","4"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is clear.""","4"
"""it would be interesting to apply the techniques of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""this reduces the results of"" AND ""in this setting, the ability to""","4"
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"""this reduces the results of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
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"""this leaves open the question of"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""a useful survey of the subject can be found in""","4"
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"""under the additional assumption that"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""under the additional assumption that"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""in future work, we plan to address questions of""","4"
"""it is well known that"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it is well known that"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""the groundbreaking work of""","4"
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"""unfortunately, we cannot assume that"" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""this leaves open the question of"" AND ""it is well known that""","4"
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"""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""recently, there has been much interest in the""","4"
"""was a major advance."" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""this reduces the results of"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""improved upon the results of""","4"
"""in this setting, the ability to"" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""who first asked whether"" AND ""we wish to extend the results of"" AND ""in this setting, the ability to""","4"
"""it has long been known that"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""in this context, the results of"" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""it would be interesting to apply the techniques of"" AND ""it is well known that"" AND ""in this context, the results of""","4"
"""the main result was the"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""this leaves open the question of""","4"
"""recently, there has been much interest in the"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""it is well known that"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""this reduces the results of"" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""unfortunately, we cannot assume that"" AND ""under the additional assumption that"" AND ""in this context, the results of""","4"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""it has long been known that""","4"
"""it is essential to consider that"" AND ""it has long been known that"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""this could shed important light on a conjecture of"" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","4"
"""it has long been known that"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""the groundbreaking work of"" AND ""in this setting, the ability to""","4"
"""recently, there has been much interest in the"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""it is well known that"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""the groundbreaking work of""","4"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""This is trivial."" AND ""This is simple."" AND ""This is obvious.""","4"
"""this leaves open the question of"" AND ""recently, there has been much interest in the"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""it is well known that"" AND ""it is essential to consider that"" AND ""a central problem in""","4"
"""this could shed important light on a conjecture of"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""was a milestone in"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","4"
"""this reduces the results of"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""in future work, we plan to address questions of""","4"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""have raised the question of whether""","4"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""every student is aware that""","4"
"""who first asked whether"" AND ""was a major advance."" AND ""in future work, we plan to address questions of""","4"
"""recently, there has been much interest in the"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""this reduces the results of"" AND ""it is essential to consider that""","4"
"""unfortunately, we cannot assume that"" AND ""it is essential to consider that"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""it is essential to consider that"" AND ""a central problem in""","4"
"""this reduces the results of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""every student is aware that""","4"
"""who first asked whether"" AND ""this leaves open the question of"" AND ""the groundbreaking work of""","4"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""This is clear.""","4"
"""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""this reduces the results of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","4"
"""this leaves open the question of"" AND ""recently, there has been much interest in the"" AND ""it is not yet known whether""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""it is well known that"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""it is essential to consider that"" AND ""a useful survey of the subject can be found in""","4"
"""it is essential to consider that"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""this reduces the results of"" AND ""this leaves open the question of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""improved upon the results of""","4"
"""it is well known that"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""it is essential to consider that"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""it is well known that"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""the groundbreaking work of"" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""in this setting, the ability to"" AND ""a useful survey of the subject can be found in"" AND ""a central problem in""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""it is not yet known whether""","4"
"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""in this context, the results of""","4"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""in this context, the results of""","4"
"""we wish to extend the results of"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""it is well known that"" AND ""it is essential to consider that"" AND ""improved upon the results of""","4"
"""in this setting, the ability to"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""under the additional assumption that"" AND ""the groundbreaking work of""","4"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""it is not yet known whether""","4"
"""this leaves open the question of"" AND ""recently, there has been much interest in the"" AND ""in this setting, the ability to""","4"
"""this could shed important light on a conjecture of"" AND ""the groundbreaking work of"" AND ""in this context, the results of""","4"
"""in this setting, the ability to"" AND ""every student is aware that"" AND ""a useful survey of the subject can be found in""","4"
"""unfortunately, we cannot assume that"" AND ""under the additional assumption that"" AND ""it is not yet known whether""","4"
"""under the additional assumption that"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""in future work, we plan to address questions of""","4"
"""this reduces the results of"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""it is not yet known whether""","4"
"""it is essential to consider that"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""in this setting, the ability to""","4"
"""was a major advance."" AND ""under the additional assumption that"" AND ""in this context, the results of""","4"
"""this leaves open the question of"" AND ""recently, there has been much interest in the"" AND ""in future work, we plan to address questions of""","4"
"""under the additional assumption that"" AND ""the groundbreaking work of"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""it is well known that""","4"
"""was a major advance."" AND ""this leaves open the question of"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is obvious.""","4"
"""in this context, the results of"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of""","4"
"""under the additional assumption that"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""was a milestone in"" AND ""was a major advance."" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""improved upon the results of""","4"
"""it is not yet known whether"" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""this could shed important light on a conjecture of"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""in future work, we plan to address questions of""","4"
"""under the additional assumption that"" AND ""it is well known that"" AND ""in this setting, the ability to""","4"
"""who first asked whether"" AND ""it is well known that"" AND ""in this context, the results of""","4"
"""we wish to extend the results of"" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""it is well known that""","4"
"""was a major advance."" AND ""under the additional assumption that"" AND ""this leaves open the question of""","4"
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"""this leaves open the question of"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""it is not yet known whether"" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""have raised the question of whether""","4"
"""was a major advance."" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
"""who first asked whether"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""in this context, the results of""","4"
"""who first asked whether"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""it would be interesting to apply the techniques of"" AND ""it is not yet known whether""","4"
"""was a major advance."" AND ""under the additional assumption that"" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""in this context, the results of""","4"
"""who first asked whether"" AND ""was a major advance."" AND ""this leaves open the question of""","4"
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"""we wish to extend the results of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of""","4"
"""under the additional assumption that"" AND ""the groundbreaking work of"" AND ""in future work, we plan to address questions of""","4"
"""under the additional assumption that"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""it is well known that"" AND ""in this context, the results of""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""in this context, the results of""","4"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""every student is aware that""","4"
"""unfortunately, we cannot assume that"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""it is essential to consider that"" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""under the additional assumption that"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""it is well known that"" AND ""a useful survey of the subject can be found in""","4"
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"""it would be interesting to apply the techniques of"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""every student is aware that""","4"
"""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of"" AND ""in this context, the results of""","4"
"""we wish to extend the results of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""the main result was the"" AND ""it is essential to consider that""","4"
"""This contradicts the fact that"" AND ""The converse is trivial."" AND ""The converse is clear.""","4"
"""this leaves open the question of"" AND ""it is well known that"" AND ""every student is aware that""","4"
"""in this setting, the ability to"" AND ""have raised the question of whether"" AND ""a useful survey of the subject can be found in""","4"
"""in this setting, the ability to"" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""this leaves open the question of"" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""every student is aware that""","4"
"""was a major advance."" AND ""this leaves open the question of"" AND ""this could shed important light on a conjecture of""","4"
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"""it would be interesting to apply the techniques of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","4"
"""the groundbreaking work of"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""who first asked whether"" AND ""was a major advance."" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
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"""this could shed important light on a conjecture of"" AND ""it is well known that"" AND ""in this context, the results of""","4"
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"""was a major advance."" AND ""the main result was the"" AND ""it is essential to consider that""","4"
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"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""a useful survey of the subject can be found in""","4"
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"""this reduces the results of"" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""in this context, the results of""","4"
"""this reduces the results of"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""every student is aware that""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""this could shed important light on a conjecture of"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""it is essential to consider that"" AND ""improved upon the results of""","4"
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"""who first asked whether"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
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"""unfortunately, we cannot assume that"" AND ""in this context, the results of"" AND ""every student is aware that""","4"
"""unfortunately, we cannot assume that"" AND ""it is well known that"" AND ""in this context, the results of""","4"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""in this context, the results of""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""this reduces the results of"" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","4"
"""This is straightforward."" AND ""This is elementary."" AND ""This is clear.""","4"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""in this context, the results of""","4"
"""we wish to extend the results of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""it would be interesting to apply the techniques of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""it is well known that""","4"
"""in this setting, the ability to"" AND ""improved upon the results of"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""it is not yet known whether""","4"
"""this leaves open the question of"" AND ""recently, there has been much interest in the"" AND ""in this context, the results of""","4"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""improved upon the results of""","4"
"""it is well known that"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""it is not yet known whether"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""every student is aware that""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""in this context, the results of""","4"
"""who first asked whether"" AND ""this leaves open the question of"" AND ""in future work, we plan to address questions of""","4"
"""the groundbreaking work of"" AND ""recently, there has been much interest in the"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""every student is aware that""","4"
"""was a major advance."" AND ""this leaves open the question of"" AND ""in this context, the results of""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""in this context, the results of""","4"
"""it is not yet known whether"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""who first asked whether"" AND ""was a major advance."" AND ""unfortunately, we cannot assume that""","4"
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"""under the additional assumption that"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""it is not yet known whether"" AND ""it has long been known that"" AND ""in this setting, the ability to""","4"
"""this could shed important light on a conjecture of"" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
"""this reduces the results of"" AND ""it is well known that"" AND ""in this setting, the ability to""","4"
"""was a major advance."" AND ""recently, there has been much interest in the"" AND ""in future work, we plan to address questions of""","4"
"""we wish to extend the results of"" AND ""this leaves open the question of"" AND ""in this setting, the ability to""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""the groundbreaking work of""","4"
"""We begin by observing that"" AND ""This is straightforward."" AND ""This is simple.""","4"
"""who first asked whether"" AND ""this leaves open the question of"" AND ""in this context, the results of""","4"
"""it is well known that"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","4"
"""this reduces the results of"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""this could shed important light on a conjecture of"" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""the groundbreaking work of""","4"
"""We proceed by induction."" AND ""This is trivial."" AND ""This is simple.""","4"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""in this setting, the ability to"" AND ""in future work, we plan to address questions of"" AND ""a central problem in""","4"
"""this reduces the results of"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","4"
"""who first asked whether"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","4"
"""under the additional assumption that"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""who first asked whether"" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""improved upon the results of""","4"
"""was a major advance."" AND ""it is essential to consider that"" AND ""have raised the question of whether""","4"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""it is essential to consider that"" AND ""a useful survey of the subject can be found in""","4"
"""in this setting, the ability to"" AND ""every student is aware that"" AND ""a central problem in""","4"
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"""unfortunately, we cannot assume that"" AND ""it would be interesting to apply the techniques of"" AND ""in this context, the results of""","4"
"""under the additional assumption that"" AND ""it is not yet known whether"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""this could shed important light on a conjecture of"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","4"
"""recently, there has been much interest in the"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""this could shed important light on a conjecture of"" AND ""in future work, we plan to address questions of""","4"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""in this context, the results of""","4"
"""this could shed important light on a conjecture of"" AND ""it is well known that"" AND ""in future work, we plan to address questions of""","4"
"""this could shed important light on a conjecture of"" AND ""recently, there has been much interest in the"" AND ""in this setting, the ability to""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it is well known that"" AND ""in future work, we plan to address questions of""","4"
"""in this setting, the ability to"" AND ""improved upon the results of"" AND ""a central problem in""","4"
"""in this setting, the ability to"" AND ""in this context, the results of"" AND ""a useful survey of the subject can be found in""","4"
"""we wish to extend the results of"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""in future work, we plan to address questions of"" AND ""every student is aware that""","4"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The converse is trivial.""","4"
"""who first asked whether"" AND ""it would be interesting to apply the techniques of"" AND ""in this setting, the ability to""","4"
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"""unfortunately, we cannot assume that"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""in future work, we plan to address questions of""","4"
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"""it is essential to consider that"" AND ""in future work, we plan to address questions of"" AND ""a central problem in""","4"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""it is well known that""","4"
"""the groundbreaking work of"" AND ""it is well known that"" AND ""it is essential to consider that""","4"
"""the groundbreaking work of"" AND ""it would be interesting to apply the techniques of"" AND ""it is essential to consider that""","4"
"""who first asked whether"" AND ""was a major advance."" AND ""in this context, the results of""","4"
"""pub.1120430586"" OR ""pub.1103368574"" OR ""pub.1121108383"" OR ""pub.1099702813""","4"
"""was a major advance."" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","4"
"""unfortunately, we cannot assume that"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""We follow"" AND ""This is straightforward."" AND ""The essential idea is that""","4"
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"""it is well known that"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
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"""this could shed important light on a conjecture of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""recently, there has been much interest in the"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""this could shed important light on a conjecture of"" AND ""it is essential to consider that""","4"
"""it is not yet known whether"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""a useful survey of the subject can be found in""","4"
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"""it is well known that"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","4"
"""it is not yet known whether"" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""it is essential to consider that"" AND ""in this context, the results of"" AND ""a central problem in""","4"
"""the groundbreaking work of"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
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"""who first asked whether"" AND ""it is well known that"" AND ""in future work, we plan to address questions of""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""it would be interesting to apply the techniques of""","4"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""this could shed important light on a conjecture of""","4"
"""was a major advance."" AND ""in this context, the results of"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""it is well known that"" AND ""in this setting, the ability to"" AND ""every student is aware that""","4"
"""in this context, the results of"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""a useful survey of the subject can be found in""","4"
"""we wish to extend the results of"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","4"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""in future work, we plan to address questions of""","4"
"""it would be interesting to apply the techniques of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","4"
"""who first asked whether"" AND ""unfortunately, we cannot assume that"" AND ""in future work, we plan to address questions of""","4"
"""the groundbreaking work of"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""recently, there has been much interest in the"" AND ""it is well known that"" AND ""in this context, the results of""","4"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""a useful survey of the subject can be found in""","4"
"""it is essential to consider that"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","4"
"""we wish to extend the results of"" AND ""was a major advance."" AND ""it is essential to consider that""","4"
"""this leaves open the question of"" AND ""it is well known that"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""the groundbreaking work of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""in this context, the results of""","4"
"""unfortunately, we cannot assume that"" AND ""recently, there has been much interest in the"" AND ""in this context, the results of""","4"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""a central problem in""","4"
"""was a major advance."" AND ""this leaves open the question of"" AND ""a useful survey of the subject can be found in""","4"
"""it is not yet known whether"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","4"
"""who first asked whether"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""in this setting, the ability to"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","4"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""a useful survey of the subject can be found in""","4"
"""was a milestone in"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","4"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""a central problem in""","4"
"""unfortunately, we cannot assume that"" AND ""this could shed important light on a conjecture of"" AND ""in future work, we plan to address questions of""","4"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""have raised the question of whether""","4"
"""We begin by observing that"" AND ""This is trivial."" AND ""This is simple.""","3"
"""was a major advance."" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","3"
"""this reduces the results of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","3"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","3"
"""We proceed by induction."" AND ""This is clear."" AND ""The essential idea is that""","3"
"""under the additional assumption that"" AND ""this leaves open the question of"" AND ""in this setting, the ability to""","3"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""this could shed important light on a conjecture of"" AND ""the groundbreaking work of"" AND ""in this setting, the ability to""","3"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""this could shed important light on a conjecture of"" AND ""it is well known that"" AND ""in this setting, the ability to""","3"
"""was a major advance."" AND ""under the additional assumption that"" AND ""in this setting, the ability to""","3"
"""this could shed important light on a conjecture of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","3"
"""who first asked whether"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""a central problem in""","3"
"""who first asked whether"" AND ""was a major advance."" AND ""in this setting, the ability to""","3"
"""in this setting, the ability to"" AND ""in future work, we plan to address questions of"" AND ""a useful survey of the subject can be found in""","3"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""in this context, the results of""","3"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is trivial.""","3"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is obvious.""","3"
"""it is well known that"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","3"
"""This is straightforward."" AND ""This is clear."" AND ""The essential idea is that""","3"
"""this leaves open the question of"" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","3"
"""We proceed by induction."" AND ""We follow"" AND ""The essential idea is that""","3"
"""unfortunately, we cannot assume that"" AND ""it is essential to consider that"" AND ""in this context, the results of""","3"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""every student is aware that""","3"
"""the groundbreaking work of"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","3"
"""was a major advance."" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","3"
"""unfortunately, we cannot assume that"" AND ""this leaves open the question of"" AND ""it is essential to consider that""","3"
"""this could shed important light on a conjecture of"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""in this context, the results of""","3"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","3"
"""under the additional assumption that"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""recently, there has been much interest in the"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is obvious.""","3"
"""this leaves open the question of"" AND ""this could shed important light on a conjecture of"" AND ""in this setting, the ability to""","3"
"""it would be interesting to apply the techniques of"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""We show the contrapositive."" AND ""This is obvious."" AND ""This is clear.""","3"
"""who first asked whether"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""unfortunately, we cannot assume that"" AND ""the groundbreaking work of"" AND ""it is essential to consider that""","3"
"""this leaves open the question of"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""who first asked whether"" AND ""this leaves open the question of"" AND ""in this setting, the ability to""","3"
"""the groundbreaking work of"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","3"
"""was a major advance."" AND ""this leaves open the question of"" AND ""it is essential to consider that""","3"
"""this could shed important light on a conjecture of"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""in this setting, the ability to"" AND ""in future work, we plan to address questions of"" AND ""improved upon the results of""","3"
"""We begin by observing that"" AND ""This is trivial."" AND ""This is straightforward.""","3"
"""it is well known that"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","3"
"""this leaves open the question of"" AND ""it is well known that"" AND ""in this setting, the ability to""","3"
"""was a major advance."" AND ""unfortunately, we cannot assume that"" AND ""it is essential to consider that""","3"
"""the groundbreaking work of"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""this leaves open the question of"" AND ""it is well known that"" AND ""it is essential to consider that""","3"
"""was a major advance."" AND ""it is essential to consider that"" AND ""in this context, the results of""","3"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","3"
"""who first asked whether"" AND ""it is well known that"" AND ""in this setting, the ability to""","3"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""this leaves open the question of"" AND ""it is not yet known whether"" AND ""in this setting, the ability to""","3"
"""it is essential to consider that"" AND ""in this context, the results of"" AND ""in future work, we plan to address questions of""","3"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""improved upon the results of""","3"
"""it is well known that"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""it is not yet known whether"" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""it is well known that"" AND ""it is essential to consider that"" AND ""in this context, the results of""","3"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""in future work, we plan to address questions of""","3"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","3"
"""recently there has been much interest in the extension of""","3"
"""was a major advance."" AND ""in this setting, the ability to"" AND ""in future work, we plan to address questions of""","3"
"""it is essential to consider that"" AND ""in this setting, the ability to"" AND ""a useful survey of the subject can be found in""","3"
"""unfortunately, we cannot assume that"" AND ""it is not yet known whether"" AND ""it is essential to consider that""","3"
"""We follow"" AND ""This is simple."" AND ""The essential idea is that""","3"
"""The converse is straightforward."" AND ""The converse is obvious."" AND ""The converse is clear.""","3"
"""This is trivial."" AND ""This is straightforward."" AND ""This is elementary.""","2"
"""We follow"" AND ""We begin by observing that"" AND ""This proof can be omitted on a first reading.""","2"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The converse is straightforward.""","2"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is straightforward.""","2"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This proof can be omitted on a first reading.""","2"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The interested reader can fill in the details.""","2"
"""was a major advance."" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","2"
"""This contradicts the fact that"" AND ""The converse is trivial."" AND ""The converse is obvious.""","2"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","2"
"""The converse is trivial."" AND ""The converse is obvious."" AND ""The converse is clear.""","2"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is clear.""","2"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","2"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This is straightforward.""","2"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""We proceed by induction.""","2"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading.""","2"
"""it is not yet known whether"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","2"
"""This is obvious."" AND ""This is clear."" AND ""The essential idea is that""","2"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This proof can be omitted on a first reading.""","2"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","2"
"""We follow"" AND ""This is trivial."" AND ""The essential idea is that""","2"
"""We begin by observing that"" AND ""This is elementary."" AND ""This is clear.""","2"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","2"
"""this leaves open the question of"" AND ""it is essential to consider that"" AND ""in this setting, the ability to""","2"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is clear.""","2"
"""We proceed by induction."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","1"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","1"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This is straightforward.""","1"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is clear.""","1"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are obvious.""","1"
"""This completes the proof."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","1"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","1"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","1"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","1"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","1"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""This is clear.""","1"
"""This completes the proof."" AND ""The result now follows by"" AND ""The converse is elementary.""","1"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","1"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""This proof can be omitted on a first reading.""","1"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","1"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","1"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","1"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""This contradicts the fact that""","1"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is straightforward, so we consider the converse.""","1"
"""We follow"" AND ""This is simple."" AND ""Suppose the contrary.""","1"
"""We follow"" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is obvious.""","1"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","1"
"""This is simple."" AND ""This is obvious."" AND ""The essential idea is that""","1"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading.""","1"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This proof can be omitted on a first reading.""","1"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are clear.""","1"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is simple.""","1"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","1"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","1"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This proof can be omitted on a first reading.""","1"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading.""","1"
"""We begin by observing that"" AND ""This is trivial."" AND ""This is elementary.""","1"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is straightforward.""","1"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","1"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This is obvious.""","1"
"""This is elementary."" AND ""This is clear."" AND ""Suppose the contrary.""","1"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""Suppose the contrary.""","1"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is simple.""","1"
"""We begin by observing that"" AND ""This is obvious."" AND ""Suppose the contrary.""","1"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is straightforward.""","1"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","1"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""Suppose the contrary.""","1"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The result now follows by""","1"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""Suppose the contrary.""","1"
"""We proceed by induction."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","1"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","1"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","1"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is simple.""","1"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","1"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is straightforward.""","1"
"""We follow"" AND ""We begin by observing that"" AND ""This is elementary.""","1"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This is simple.""","1"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","1"
"""We proceed by induction."" AND ""We follow"" AND ""This proof can be omitted on a first reading.""","1"
"""We show the contrapositive."" AND ""This is simple."" AND ""This is clear.""","1"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""We begin by observing that""","1"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""We begin by observing that""","1"
"""This is a contradiction."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","1"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","1"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","1"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""Suppose the contrary.""","1"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","1"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","1"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is clear.""","1"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are clear.""","1"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is straightforward.""","1"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""We begin by considering a simple special case.""","1"
"""This is straightforward."" AND ""This is simple."" AND ""This is elementary.""","1"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","1"
"""This is a contradiction."" AND ""The converse is straightforward."" AND ""The converse is clear.""","1"
"""We proceed by induction."" AND ""This is trivial."" AND ""This is elementary.""","1"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""Suppose the contrary.""","1"
"""We proceed by induction."" AND ""This is simple."" AND ""Suppose the contrary.""","1"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","1"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is straightforward.""","1"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""This is simple."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","1"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""Suppose the contrary.""","1"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","1"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""This is clear.""","1"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""This is clear.""","1"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are clear.""","1"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""This contradicts the fact that""","1"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is simple.""","1"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""We follow""","1"
"""We proceed by induction."" AND ""This is simple."" AND ""This is obvious.""","1"
"""This is straightforward."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","1"
"""This is obvious."" AND ""This is elementary."" AND ""Suppose the contrary.""","1"
"""We begin by observing that"" AND ""This is elementary."" AND ""Suppose the contrary.""","1"
"""We show the contrapositive."" AND ""We follow"" AND ""This proof can be omitted on a first reading.""","1"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","1"
"""We begin by observing that"" AND ""This is straightforward."" AND ""This is elementary.""","1"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The interested reader can fill in the details.""","1"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""We begin by observing that""","1"
"""This is trivial."" AND ""This is simple."" AND ""This is elementary.""","1"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""This is clear.""","1"
"""We begin by observing that"" AND ""This is obvious."" AND ""This is elementary.""","1"
"""This is trivial."" AND ""This is straightforward."" AND ""The essential idea is that""","1"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is simple.""","1"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is trivial.""","1"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""This is clear.""","1"
"""We begin by observing that"" AND ""This is clear."" AND ""Suppose the contrary.""","1"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are obvious.""","1"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is clear.""","1"
"""This contradicts the fact that"" AND ""The converse is obvious."" AND ""The converse is clear.""","1"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is straightforward.""","1"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","1"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is clear.""","1"
"""We begin by observing that"" AND ""This is trivial."" AND ""This is obvious.""","1"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The interested reader can fill in the details.""","1"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""This is clear.""","1"
"""We follow"" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","1"
"""We follow"" AND ""We begin by observing that"" AND ""We begin by considering a simple special case.""","1"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","1"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This is trivial.""","1"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is trivial.""","1"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This proof can be omitted on a first reading.""","1"
"""We follow"" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","1"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","1"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are obvious.""","1"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is clear.""","1"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""The converse is straightforward."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is simple.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are simple.""","0"
"""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""The converse is trivial."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are obvious."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are simple.""","0"
"""This is clear."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is trivial.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""This clearly implies the result.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""We begin by considering a simple special case.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""This clearly implies the result.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""The interested reader can fill in the details."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""This is obvious.""","0"
"""The remaining details are simple."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""We begin by considering a simple special case.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""The converse is straightforward."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is clear."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""This is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is straightforward.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is elementary.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This is simple."" AND ""This is obvious."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""This is obvious."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is simple."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is elementary.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""This is the desired statement."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is simple."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""The converse is trivial."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are elementary."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""We follow"" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The result now follows by""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is trivial.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","0"
"""We proceed by induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is straightforward.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The converse is straightforward."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""The remaining details are elementary."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is clear.""","0"
"""This is a contradiction."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are trivial.""","0"
"""The remaining details are clear."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""This is clear."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are trivial.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""The result now follows by"" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This is elementary.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is simple.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is elementary.""","0"
"""The converse is straightforward."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""The converse is trivial."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is straightforward.""","0"
"""This clearly implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is clear, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is obvious.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""We begin by considering a simple special case.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""The essential idea is that""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are simple.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is clear, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""This is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is elementary.""","0"
"""The remaining details are trivial."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""The interested reader can fill in the details."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""This is a contradiction.""","0"
"""This is simple."" AND ""This is elementary."" AND ""This is clear.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is trivial.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""The interested reader can fill in the details."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""This completes the proof.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""Suppose the contrary.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is clear.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""This is clear.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""The remaining details are straightforward."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""We proceed by transfinite induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is obvious.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""We proceed by induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The converse is obvious."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""We begin by observing that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""This contradicts the fact that""","0"
"""The remaining details are trivial."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The interested reader can fill in the details."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is straightforward.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""This completes the proof.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""This is clear."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""Suppose the contrary.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""This contradicts the fact that""","0"
"""This is a contradiction."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is trivial.""","0"
"""The interested reader can fill in the details."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""This clearly implies the result."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""This clearly implies the result.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is elementary.""","0"
"""The remaining details are simple."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""Suppose the contrary.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are obvious.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""We follow"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""This is trivial."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is obvious.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""This is obvious.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""The converse is simple."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""The remaining details are simple."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""The essential idea is that""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is obvious.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are elementary.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""The remaining details are simple."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are straightforward.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is clear.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""This is simple."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are simple.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are clear.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""The result now follows by"" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is straightforward.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""This clearly implies the result."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""We begin by considering a simple special case.""","0"
"""This is the desired statement."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The converse is obvious."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""This is clear.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""The remaining details are straightforward."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""The essential idea is that""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The result now follows by"" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""This is obvious.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are clear.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""The converse is straightforward."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""The remaining details are simple."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The converse is elementary.""","0"
"""This is obvious."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""This is elementary.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are straightforward.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""This is the desired statement."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""The converse is straightforward."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is straightforward.""","0"
"""The result now follows by"" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""This is clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is obvious.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The interested reader can fill in the details.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""This is obvious.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""This contradicts the fact that"" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""The remaining details are straightforward."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This is obvious."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""This is clear.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This trivially implies the result."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""The remaining details are clear."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is clear.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""The essential idea is that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are obvious.""","0"
"""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""This is simple."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is simple."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""The result now follows by"" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""This is simple."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This clearly implies the result."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The result now follows by""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""This clearly implies the result.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""Suppose the contrary.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""This clearly implies the result.""","0"
"""This is clear."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""Suppose the contrary.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The interested reader can fill in the details.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are simple.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is trivial.""","0"
"""This is a contradiction."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""This is elementary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is clear."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The converse is trivial."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is straightforward.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are clear.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""This is obvious.""","0"
"""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""This is a contradiction.""","0"
"""This is obvious."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is obvious.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""The remaining details are obvious."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is obvious.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is clear.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is left as an exercise to the reader.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is elementary.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The interested reader can fill in the details.""","0"
"""The converse is trivial."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The result now follows by""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""This is simple.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""Suppose the contrary.""","0"
"""The remaining details are clear."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""Suppose the contrary.""","0"
"""The converse is obvious."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are trivial."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are clear.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""The essential idea is that""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is trivial.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are simple.""","0"
"""The remaining details are clear."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are clear.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""This is simple.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are elementary.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The remaining details are obvious.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are straightforward.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""The remaining details are trivial."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are clear."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""This is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This clearly implies the result."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are clear.""","0"
"""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is obvious.""","0"
"""The remaining details are clear."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is elementary.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is clear.""","0"
"""This completes the proof."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are elementary.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""The converse is trivial."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""This clearly implies the result.""","0"
"""The remaining details are clear."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""This is obvious.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""The result now follows by"" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This is trivial."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is trivial.""","0"
"""The remaining details are simple."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""The essential idea is that""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""Suppose the contrary.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The result now follows by""","0"
"""This is elementary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is trivial.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are trivial.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""The converse is trivial."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""Suppose the contrary.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are simple.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The result now follows by""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is trivial.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""This completes the proof."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The result now follows by""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""This is elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is clear.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is simple.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The result now follows by""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""We begin by considering a simple special case.""","0"
"""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""This is elementary.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""This clearly implies the result.""","0"
"""The converse is simple."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""The converse is trivial."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""Suppose the contrary.""","0"
"""This is simple."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is obvious.""","0"
"""The remaining details are elementary."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""The interested reader can fill in the details."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This completes the proof."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""We begin by observing that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""This clearly implies the result.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""This clearly implies the result.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are clear.""","0"
"""This is elementary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The converse is trivial."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""This is elementary.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is elementary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is trivial.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""Suppose the contrary.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""This clearly implies the result.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This is elementary."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are trivial.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""This is elementary.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is obvious."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This is clear."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""This is the desired statement."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""This is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is simple.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""The converse is trivial."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""This completes the proof.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are elementary.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""This is simple.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""The remaining details are obvious."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""This completes the proof.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""Suppose the contrary.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is straightforward."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""The essential idea is that""","0"
"""The remaining details are obvious."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are obvious.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""The converse is simple."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""This completes the proof.""","0"
"""The remaining details are simple."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""Suppose the contrary.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are trivial.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is straightforward.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are simple.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is obvious."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""Suppose the contrary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading.""","0"
"""This is a contradiction."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""This contradicts the fact that""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are trivial.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""The remaining details are elementary."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are straightforward.""","0"
"""We begin by observing that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""This contradicts the fact that""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is obvious."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""We follow"" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""This is elementary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is obvious.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are obvious.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The result now follows by""","0"
"""The result now follows by"" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is straightforward.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by observing that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is elementary.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""Suppose the contrary.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""This is elementary.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are straightforward.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""This is elementary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are clear.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""We begin by considering a simple special case.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is simple.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The converse is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""This clearly implies the result.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""This is obvious.""","0"
"""The remaining details are elementary."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""The converse is straightforward."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""This is elementary.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is clear, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are obvious.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are straightforward.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is trivial."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""This is clear.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The interested reader can fill in the details.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are trivial.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""This is simple."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""This is obvious.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are trivial."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""The essential idea is that""","0"
"""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are straightforward.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are clear."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""This is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""This completes the proof.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The interested reader can fill in the details."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""This is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""This is the desired statement.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This is straightforward."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is elementary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The result now follows by""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is trivial.""","0"
"""We follow"" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""This is elementary.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""Suppose the contrary.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The interested reader can fill in the details.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is obvious."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""This clearly implies the result.""","0"
"""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""This clearly implies the result.""","0"
"""We proceed by transfinite induction."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This is elementary."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This contradicts the fact that"" AND ""The remaining details are simple.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""This is elementary.""","0"
"""We proceed by induction."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is trivial.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is clear, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is trivial.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""This is elementary."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The remaining details are obvious.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is simple.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are trivial.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""The remaining details are simple."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""This clearly implies the result."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""We follow"" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are elementary."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This is elementary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""This is elementary.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are trivial.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The remaining details are obvious.""","0"
"""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is obvious.""","0"
"""This completes the proof."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The interested reader can fill in the details.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""We show the contrapositive."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""This is simple.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""This is straightforward."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""The essential idea is that""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""The remaining details are trivial."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""The remaining details are clear."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The essential idea is that"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is straightforward.""","0"
"""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This is straightforward."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We follow"" AND ""This is simple."" AND ""This is elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""This is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""Suppose the contrary.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""The essential idea is that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""This is a contradiction.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The converse is simple."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is trivial."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are trivial."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""We begin by considering a simple special case.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""Suppose the contrary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The remaining details are clear.""","0"
"""The remaining details are trivial."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This completes the proof."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""This obviously implies the result."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""The remaining details are straightforward."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"
"""The remaining details are obvious."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""The remaining details are clear."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The remaining details are elementary."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""This completes the proof.""","0"
"""This is a contradiction."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are trivial.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""This is elementary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""We follow"" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is trivial.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""This clearly implies the result.""","0"
"""The result now follows by"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are simple.""","0"
"""This is the desired statement."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We follow"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""The essential idea is that""","0"
"""The remaining details are trivial."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is clear, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""The interested reader can fill in the details."" AND ""The converse is straightforward."" AND ""The converse is clear.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are clear.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is simple.""","0"
"""This is trivial."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is obvious.""","0"
"""This is left as an exercise to the reader."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is trivial."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""We follow"" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""The remaining details are clear."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are trivial.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The converse is trivial.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""This is simple.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The interested reader can fill in the details.""","0"
"""This is simple."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""The converse is trivial."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This is a contradiction."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is simple.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""The converse is left as an exercise to the reader."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""This is straightforward.""","0"
"""The remaining details are trivial."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are clear.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is simple.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""This completes the proof.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This is simple.""","0"
"""The remaining details are obvious."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""This is straightforward."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The converse is simple."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is simple, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The remaining details are straightforward.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are simple.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The remaining details are simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The remaining details are obvious.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""The essential idea is that""","0"
"""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""Suppose the contrary.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The interested reader can fill in the details.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""This is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""This is simple."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""The remaining details are elementary."" AND ""The converse is straightforward."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We begin by considering a simple special case."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are elementary.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""This is elementary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The converse is simple.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","0"
"""This proof can be omitted on a first reading."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""This is left as an exercise to the reader.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""We begin by observing that"" AND ""We begin by considering a simple special case."" AND ""The essential idea is that""","0"
"""The result now follows by"" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This is simple."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The result now follows by"" AND ""The remaining details are obvious.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This obviously implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is trivial.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is obvious."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""The essential idea is that""","0"
"""This obviously implies the result."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The converse is trivial.""","0"
"""We follow"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are elementary."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The essential idea is that"" AND ""Suppose the contrary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This completes the proof."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""One direction is clear, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The remaining details are straightforward.""","0"
"""This is obvious."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are trivial.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is simple.""","0"
"""This clearly implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is simple, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is simple.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is clear.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""Suppose the contrary.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are clear."" AND ""The converse is obvious.""","0"
"""This is trivial."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""This obviously implies the result."" AND ""This contradicts the fact that"" AND ""The converse is clear.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""This is elementary."" AND ""The essential idea is that""","0"
"""This is simple."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are obvious.""","0"
"""The interested reader can fill in the details."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""The converse is simple."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is elementary.""","0"
"""This clearly implies the result."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""The remaining details are clear."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is simple.""","0"
"""This contradicts the fact that"" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""Suppose the contrary.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The converse is elementary.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The interested reader can fill in the details.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""We proceed by transfinite induction."" AND ""This is obvious."" AND ""The essential idea is that""","0"
"""This completes the proof."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are simple."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This contradicts the fact that"" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""The remaining details are simple."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are trivial.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""Suppose the contrary.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""This is elementary.""","0"
"""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is obvious."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by observing that"" AND ""This is left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are simple.""","0"
"""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are obvious."" AND ""The converse is obvious.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""This completes the proof."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The converse is trivial.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The converse is elementary.""","0"
"""This is simple."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""This is straightforward."" AND ""One direction is trivial, so we consider the converse.""","0"
"""Suppose the contrary."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""The remaining details are obvious."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is clear."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""This is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""This is simple."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is elementary."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""The essential idea is that""","0"
"""This is a contradiction."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This clearly implies the result."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is trivial."" AND ""This is elementary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is simple.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is elementary.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are trivial."" AND ""The remaining details are clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""We follow"" AND ""We begin by observing that"" AND ""The essential idea is that""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are elementary.""","0"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""The result now follows by"" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This obviously implies the result."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""The result now follows by"" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The converse is trivial.""","0"
"""We proceed by induction."" AND ""Suppose the contrary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are elementary.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""The essential idea is that"" AND ""Suppose the contrary.""","0"
"""This is trivial."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""The essential idea is that"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""This is obvious.""","0"
"""We follow"" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""We follow"" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The converse is straightforward.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader.""","0"
"""This is straightforward."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This completes the proof."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is simple.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This obviously implies the result."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is a contradiction."" AND ""The remaining details are straightforward."" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""This is straightforward."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is obvious."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The converse is obvious.""","0"
"""This is simple."" AND ""Suppose the contrary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is clear.""","0"
"""The result now follows by"" AND ""The remaining details are trivial."" AND ""The converse is trivial.""","0"
"""The essential idea is that"" AND ""One direction is elementary, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""We follow"" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are elementary."" AND ""The remaining details are clear.""","0"
"""This is the desired statement."" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""One direction is obvious, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is elementary.""","0"
"""The remaining details are clear."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This is obvious."" AND ""The essential idea is that"" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are elementary.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This is the desired statement."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""This is clear."" AND ""The essential idea is that""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""We proceed by induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The remaining details are obvious.""","0"
"""This is simple."" AND ""This is obvious."" AND ""Suppose the contrary.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is simple.""","0"
"""This is elementary."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This is straightforward."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is straightforward.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is elementary."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is simple."" AND ""This is elementary.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""This clearly implies the result."" AND ""The converse is elementary.""","0"
"""This is straightforward."" AND ""This is left as an exercise to the reader."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is trivial.""","0"
"""The remaining details are straightforward."" AND ""The converse is straightforward."" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is simple.""","0"
"""The remaining details are straightforward."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""This completes the proof."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""We follow"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This is straightforward."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The interested reader can fill in the details.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""This trivially implies the result."" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""This is a contradiction."" AND ""The remaining details are simple."" AND ""The converse is clear.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is simple, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This completes the proof."" AND ""The remaining details are straightforward.""","0"
"""This is simple."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is trivial."" AND ""This is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The remaining details are obvious.""","0"
"""The remaining details are simple."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""The result now follows by"" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is simple.""","0"
"""This is the desired statement."" AND ""This is a contradiction."" AND ""The remaining details are obvious.""","0"
"""This is a contradiction."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is simple, so we consider the converse.""","0"
"""The converse is trivial."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""This is clear.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The interested reader can fill in the details.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The remaining details are straightforward.""","0"
"""This obviously implies the result."" AND ""This is the desired statement."" AND ""The remaining details are clear.""","0"
"""This contradicts the fact that"" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""The remaining details are obvious."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are obvious.""","0"
"""We follow"" AND ""The essential idea is that"" AND ""One direction is trivial, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""Suppose the contrary."" AND ""One direction is simple, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This clearly implies the result."" AND ""The remaining details are elementary."" AND ""The converse is clear.""","0"
"""This is trivial."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The converse is simple."" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is left as an exercise to the reader.""","0"
"""We follow"" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is obvious."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is elementary."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We follow"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This completes the proof."" AND ""The remaining details are clear."" AND ""The converse is left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The converse is simple.""","0"
"""This clearly implies the result."" AND ""The remaining details are obvious."" AND ""The converse is trivial.""","0"
"""This is simple."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""The converse is trivial."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The remaining details are obvious."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is clear.""","0"
"""We proceed by transfinite induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is straightforward."" AND ""This is simple."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is simple."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are straightforward.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The result now follows by""","0"
"""We begin by considering a simple special case."" AND ""This is obvious."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""The essential idea is that"" AND ""One direction is clear, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading.""","0"
"""This is the desired statement."" AND ""The converse is trivial."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""This trivially implies the result."" AND ""This obviously implies the result."" AND ""The result now follows by""","0"
"""We begin by considering a simple special case."" AND ""This is straightforward."" AND ""This is obvious.""","0"
"""We show the contrapositive."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""One direction is clear, so we consider the converse.""","0"
"""This is the desired statement."" AND ""The remaining details are straightforward."" AND ""The remaining details are clear.""","0"
"""We proceed by induction."" AND ""We follow"" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is simple."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""This is simple.""","0"
"""This is trivial."" AND ""This is left as an exercise to the reader."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The interested reader can fill in the details."" AND ""The converse is trivial."" AND ""The converse is straightforward.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""The remaining details are straightforward."" AND ""The converse is trivial."" AND ""The converse is clear.""","0"
"""This contradicts the fact that"" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This trivially implies the result."" AND ""The remaining details are simple."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This completes the proof."" AND ""The converse is straightforward."" AND ""The converse is simple.""","0"
"""This is a contradiction."" AND ""This clearly implies the result."" AND ""The converse is obvious.""","0"
"""This is a contradiction."" AND ""The converse is simple."" AND ""The converse is clear.""","0"
"""This is straightforward."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is obvious."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The converse is straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are elementary.""","0"
"""This is the desired statement."" AND ""The remaining details are elementary."" AND ""The converse is straightforward.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are simple."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is clear.""","0"
"""We proceed by induction."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""This completes the proof."" AND ""The remaining details are clear.""","0"
"""This trivially implies the result."" AND ""This is a contradiction."" AND ""The converse is simple.""","0"
"""We begin by considering a simple special case."" AND ""This proof can be omitted on a first reading."" AND ""This is clear.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The remaining details are simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are clear."" AND ""The converse is trivial.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is simple."" AND ""This is left as an exercise to the reader.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""This is trivial.""","0"
"""This completes the proof."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is clear.""","0"
"""This clearly implies the result."" AND ""The result now follows by"" AND ""The converse is trivial.""","0"
"""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The interested reader can fill in the details.""","0"
"""We begin by observing that"" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We proceed by transfinite induction."" AND ""This is obvious.""","0"
"""This trivially implies the result."" AND ""The interested reader can fill in the details."" AND ""The converse is left as an exercise to the reader.""","0"
"""We show the contrapositive."" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader."" AND ""This is elementary.""","0"
"""We show the contrapositive."" AND ""Suppose the contrary."" AND ""One direction is clear, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""We show the contrapositive."" AND ""This is straightforward."" AND ""Suppose the contrary.""","0"
"""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""We begin by considering a simple special case."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""The essential idea is that""","0"
"""This is straightforward."" AND ""This is elementary."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The result now follows by"" AND ""The remaining details are straightforward.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The remaining details are obvious.""","0"
"""This completes the proof."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""We proceed by transfinite induction."" AND ""This is straightforward."" AND ""One direction is clear, so we consider the converse.""","0"
"""The essential idea is that"" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""We show the contrapositive."" AND ""We begin by observing that"" AND ""Suppose the contrary.""","0"
"""We follow"" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This is simple."" AND ""One direction is simple, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We follow"" AND ""This proof can be omitted on a first reading."" AND ""This is trivial.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is elementary.""","0"
"""The converse is straightforward."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""This clearly implies the result."" AND ""The result now follows by""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""The converse is straightforward."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is obvious.""","0"
"""This trivially implies the result."" AND ""The remaining details are clear."" AND ""The converse is simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are straightforward."" AND ""The remaining details are elementary.""","0"
"""This clearly implies the result."" AND ""The converse is simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are simple."" AND ""The converse is trivial."" AND ""The converse is simple.""","0"
"""We show the contrapositive."" AND ""We proceed by induction."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader."" AND ""The converse is elementary.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is trivial, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""Suppose the contrary.""","0"
"""This completes the proof."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""This proof can be omitted on a first reading."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by considering a simple special case."" AND ""This is elementary."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""We proceed by induction."" AND ""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The converse is obvious.""","0"
"""This contradicts the fact that"" AND ""This clearly implies the result."" AND ""The remaining details are clear.""","0"
"""We show the contrapositive."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by transfinite induction."" AND ""This is left as an exercise to the reader."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are simple."" AND ""The converse is elementary.""","0"
"""This is obvious."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is left as an exercise to the reader."" AND ""This is clear.""","0"
"""This is trivial."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""The result now follows by"" AND ""The remaining details are straightforward."" AND ""The converse is straightforward.""","0"
"""The remaining details are straightforward."" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""We begin by observing that"" AND ""This is straightforward."" AND ""The essential idea is that""","0"
"""This is obvious."" AND ""One direction is straightforward, so we consider the converse."" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The converse is trivial."" AND ""The converse is left as an exercise to the reader.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""Suppose the contrary.""","0"
"""This obviously implies the result."" AND ""The remaining details are elementary."" AND ""The converse is elementary.""","0"
"""We begin by considering a simple special case."" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""One direction is trivial, so we consider the converse."" AND ""One direction is obvious, so we consider the converse."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""This completes the proof."" AND ""The converse is left as an exercise to the reader.""","0"
"""The remaining details are straightforward."" AND ""The remaining details are elementary."" AND ""The converse is obvious.""","0"
"""The remaining details are left as an exercise to the reader."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This proof can be omitted on a first reading."" AND ""This is straightforward."" AND ""This is simple.""","0"
"""The remaining details are trivial."" AND ""The remaining details are elementary."" AND ""The interested reader can fill in the details.""","0"
"""This obviously implies the result."" AND ""This is a contradiction."" AND ""The remaining details are trivial.""","0"
"""This is clear."" AND ""One direction is trivial, so we consider the converse."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""The remaining details are trivial."" AND ""The converse is clear.""","0"
"""We begin by considering a simple special case."" AND ""One direction is left as an exercise to the reader, so we consider the converse."" AND ""One direction is clear, so we consider the converse.""","0"
"""The remaining details are clear."" AND ""The interested reader can fill in the details."" AND ""The converse is trivial.""","0"
"""The result now follows by"" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This clearly implies the result."" AND ""The remaining details are trivial."" AND ""The remaining details are left as an exercise to the reader.""","0"
"""This contradicts the fact that"" AND ""The result now follows by"" AND ""The converse is left as an exercise to the reader.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""This clearly implies the result.""","0"
"""This contradicts the fact that"" AND ""The remaining details are simple."" AND ""The remaining details are elementary.""","0"
"""This clearly implies the result."" AND ""The converse is obvious."" AND ""The converse is elementary.""","0"
"""This is left as an exercise to the reader."" AND ""This is clear."" AND ""Suppose the contrary.""","0"
"""We show the contrapositive."" AND ""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse.""","0"
"""We show the contrapositive."" AND ""We follow"" AND ""One direction is simple, so we consider the converse.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is trivial."" AND ""This is simple."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is trivial."" AND ""One direction is trivial, so we consider the converse.""","0"
"""This is trivial."" AND ""The essential idea is that"" AND ""One direction is obvious, so we consider the converse.""","0"
"""The remaining details are elementary."" AND ""The interested reader can fill in the details."" AND ""The converse is straightforward.""","0"
"""This completes the proof."" AND ""The result now follows by"" AND ""The converse is straightforward.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""The essential idea is that""","0"
"""The remaining details are simple."" AND ""The converse is simple."" AND ""The converse is elementary.""","0"
"""This is clear."" AND ""Suppose the contrary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This clearly implies the result."" AND ""The remaining details are left as an exercise to the reader."" AND ""The remaining details are clear.""","0"
"""We show the contrapositive."" AND ""This proof can be omitted on a first reading."" AND ""This is left as an exercise to the reader.""","0"
"""We follow"" AND ""We begin by considering a simple special case."" AND ""One direction is simple, so we consider the converse.""","0"
"""This is elementary."" AND ""This is clear."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We begin by observing that"" AND ""This is elementary."" AND ""One direction is simple, so we consider the converse.""","0"
"""We proceed by induction."" AND ""This is elementary."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This obviously implies the result."" AND ""The result now follows by"" AND ""The converse is simple.""","0"
"""This is straightforward."" AND ""Suppose the contrary."" AND ""One direction is obvious, so we consider the converse.""","0"
"""We proceed by induction."" AND ""We begin by observing that"" AND ""This is left as an exercise to the reader.""","0"
"""We proceed by transfinite induction."" AND ""We proceed by induction."" AND ""We begin by considering a simple special case.""","0"
"""We follow"" AND ""This is trivial."" AND ""One direction is elementary, so we consider the converse.""","0"
"""This contradicts the fact that"" AND ""The interested reader can fill in the details."" AND ""The converse is obvious.""","0"
"""This proof can be omitted on a first reading."" AND ""This is clear."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This proof can be omitted on a first reading."" AND ""This is obvious."" AND ""This is left as an exercise to the reader.""","0"
"""We begin by observing that"" AND ""This is simple."" AND ""One direction is simple, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This is the desired statement."" AND ""The converse is obvious.""","0"
"""We proceed by induction."" AND ""This is obvious."" AND ""One direction is straightforward, so we consider the converse.""","0"
"""This trivially implies the result."" AND ""This contradicts the fact that"" AND ""The remaining details are trivial.""","0"
"""This contradicts the fact that"" AND ""The converse is trivial."" AND ""The converse is elementary.""","0"
"""This proof can be omitted on a first reading."" AND ""This is elementary."" AND ""One direction is left as an exercise to the reader, so we consider the converse.""","0"
"""This is the desired statement."" AND ""This contradicts the fact that"" AND ""The converse is straightforward.""","0"
"""This is the desired statement."" AND ""The remaining details are simple."" AND ""The converse is left as an exercise to the reader.""","0"
"""This is a contradiction."" AND ""The remaining details are obvious."" AND ""The converse is clear.""","0"
"""This is the desired statement."" AND ""The remaining details are left as an exercise to the reader."" AND ""The converse is obvious.""","0"