# 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)
.