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Section 1.7: Problem 6 Solution

Working problems is a crucial part of learning mathematics. No one can learn... merely by poring over the definitions, theorems, and examples that are worked out in the text. One must work part of it out for oneself. To provide that opportunity is the purpose of the exercises.
James R. Munkres
Let be a deduction from a set of wffs, as in the preceding problem. Show that for each . Suggestion: Use strong induction on k, so that the inductive hypothesis is that for all .
Let , i.e. the set of the first wffs forming the deduction. We show that if then for each , and, in particular, . Since , . Suppose that . Then if is a tautology, then , if , then , and, finally, if is such that for some , then implying (Exercise 4(a), Section 1.2) .