| NOTE 3: In the above "skeleton" of PROOF OF (*), N denotes some integer greater than or equal to 2. STEPS 2 through N are all devoted
towards reaching the whole goal of PROOF OF (*), STEP N + 1. Having finished STEP N + 1, we have completed PROOF OF (*), since, at this point,
it has been shown that the HYPOTHESIS of (*) implies the CONCLUSION of (*) (i.e. it has been shown that (*) is true). |
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| NOTE 4: Once a conditional statement, (*), has been proven, we are free to use it in a proof of any other statement, (**). Of course,
we would never do this unless it helps to prove (**). After all, we would never waste our time discussing an irrelevant conditional statement, (*).
But, when we want to do this, we proceed, after any given step, STEP K, in A PROOF OF (**), at which we are able to successfully complete STEP K
+ 1 in THIS PROOF OF (**), as follows : |
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USE OF A PREVIOUSLY PROVEN CONDITIONAL STATEMENT, (*), IN A
PROOF OF ANOTHER CONDITIONAL STATEMENT, (**): |
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STEP K + 1 OF THIS PROOF OF (**): Deduce the HYPOTHESIS of (*). |
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STEP K + 2 OF THIS PROOF OF (**): Appeal to the previously proven conditional statement, (*), to conclude that, since the conditional statement
(*) holds, as previously proven, and the HYPOTHESIS of (*) holds, as proven in STEP K + 1 OF THIS PROOF OF (**), the CONCLUSION OF (*) also holds. |
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| REMARK 5: Once STEP K + 2 OF THIS PROOF OF (**) is completed, we may now use the CONCLUSION OF (*) at any appropriate subsequent step of THIS
PROOF OF (*). |
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| REMARK 6: Note that a conditional statement (*) asserts nothing about whether or not the CONCLUSION of (*) is true in any situation in which the
HYPOTHESIS of (*) is not true. It only gives us information about whether or not the CONCLUSION of (*) is true in situations in which the HYPOTHESIS OF
(*) is true. More precisely, in any of the situations in which the HYPOTHESIS OF (*) is true, the conditional statement, (*), asserts that the CONCLUSION
OF (*) is also true. |
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| REMARK 7: Of course, if a conditional statement (*) is not true, then its assertion is of no use in any proof of any statement. This issue is, of
course, of real concern to us. We do not want to be building a "proof" of some statement, (**), on any conditional statements, (*), of dubious validity.
Such a "proof" of (**) is not a proof of (**) at all. Once a conditional statement, (*), is proven, however, this issue is of no further concern to us. We
may use such a previously proven conditional statement at any point at which we find it useful to do so. |
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| REMARK 8: Of course, even if a conditional statement, (*), has been previously proven, it will not be useful to us in building a proof
of some statement, (**), unless it is relevant to the discussion of this proof of (**) at the point at which we plan to use it. This question of the
relevance of (*) at this point of this proof of (**) is answered affirmatively (i.e. "Yes, (*) is relevant at this point of this proof of (**).) when
STEP K + 1 above is successfully completed. |
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| REMARK 9: STEP K + 2 accomplishes our sole purpose in using (*). Our precise purpose in using (*) was to get the CONCLUSION OF (*). Our plan for
proving (**) involved using the CONCLUSION OF (*). But, since we were not given the CONCLUSION OF (*), we had to find some way to get the CONCLUSION OF
(*). Knowing the previously proven result, (*), and being confident that it would be "easier" to first prove the HYPOTHESIS OF (*), rather than to try to
prove the CONCLUSION OF (*) "directly", we decided to take an "indirect route" to the CONCLUSION OF (*). Namely, we decided to first prove the HYPOTHESIS
OF (*), then appeal to the previously proven result, (*), to get the CONCLUSION OF (*). |
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| ADVICE 10: Work on developing the skill of recognizing conditional statements even when they are "hidden" (i.e. when they are expressed in a
logically equivalent way other than the skeletal form of a conditional statement : If HYPOTHESIS, then CONCLUSION). |