I'm seeking for a proof of Theorem 9.2.12. in the Gompf-Stipsicz "*4-Manifolds and Kirby Calculus*" (for the statement, see the following image). But the textbook omits any proofs and only gives a weaker version (Theorem 9.2.13.). My question is:

**How can we find a proof of the theorem?**

Any references will be fine. Thank you for your help.

The Wide World of 4-Manifoldsby A. Scorpan (google books) for a recent exposition of the Wall reference and its corollaries (p. 155, linked). $\endgroup$