# Section 3.3: Problem 4 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

Show that in the proof preceding Theorem 33L, formula (4) is logically implied by the set consisting of formulas (1), (2), and (3).

Let us recall that we show that
is representable if
,
and
are (functionally) representable (by
,
and
, respectively). We let
be
, and argue that for every
,
imply

To show (4), by the generalization (DG) theorem, it is sufficient to show
.

Now,
is logically equivalent to
(the prenex normal form, see Section 2.6), and to show
it is sufficient to show that
for some terms
and
substitutable for
and
, respectively. So, let
. Then, by the deduction (DD) and contraposition (DC) theorems, it is sufficient to show that
. By using deduction axiom group 2 (DA2), we can get rid of quantifiers in equations (1)-(3) and substitute
for
, hence,
Equation (5), by DA1 and modus ponens (DMP), implies
, and equations (6) and (7), by DA5, DA2, DA1 and DMP, imply
and
.

To show
, by DD and DG, it is sufficient to show that
. From (5), by DA1 and DMP,
. Further, by DA2, DA1 and DMP, equations (2) and (3) imply
and
, respectively. Moreover, by DA6,
and
, from which, by using DA2 and substituting
for
and
for
, we obtain
and
. Overall, by DMP, we have
.