Could you check if my following attempt is O.K ?

Since every recursive relation is representable in the theory by assumption, there is a formula represents corresponding to in the theory (Enderton 2nd edition, pp 185).

Let be the Gödel number of .

Let be .

Then, says that no number is the value of (Enderton 2nd edition, pp 185) at a deduction of .

If , there is a contradiction explained in the textbook (Enderton 2nd edition, pp 185).

Therefore, we have .

Now, we need to show that .

Suppose to the contrary that .

Then, .

By -consistency, it cannot be for all . It follows that it cannot be for every . Now, . It means is false in , contradicting that is true in (Enderton 2nd edition, pp 185).

Thus .