My question is about the paper "Generic automorphisms of fields" by A. Macintyre. (It is about the model theory of difference fields.) In section 1.5, after the proof of Lemma 11, he makes the following remark:
$$\text{From this, Lemma 8 and the standard criterion [10] for quantifier-elimination, we conclude...}$$
By [10], he refers to the book "Saturated Model Theory" by G. Sacks. My question: What exactly is the "standard criterion for quantifier-elimination"? Sack's book contains several criteria for the elimination of quantifiers but I didn't found one which he calls the "standard criterion". Does anyone knows what Lemma or Theorem from Sack's book Macintyre might have in mind? Thanks for help!
Edit: Explaining the whole context of Macintyre's remark would be quite tedious. So, my question is addressed only to people who might know the paper...
I believe the criterion Macintyre is referring to is Theorem 13.1 of Saturated Model Theory. Paraphrasing, this theorem says that $T$ has quantifier-elimination if and only if whenever $f\colon A\to B$ and $g\colon A\to C$ are embeddings, where $A$ is an $L$-structure and $B$ and $C$ are models of $T$, there is a structure $D$ and embeddings $f'\colon B\to D$ and $g'\colon C\to D$ such that $f'$ is an elementary embedding, and such that $f'\circ f = g'\circ g$.
Note that if we already know $T$ is model-complete, then as long as $D\models T$, the condition that $f'$ is an elementary embedding is automatic.
Now Macintyre wants to apply this to show that the theory $T$ of e.c. difference fields, considered in the language $L$ extending the language of difference fields with the extra definable predicates $S_{f,\overline{g},\overline{h}}$, has quantifier elimination. Since $T$ is a theory of e.c. structures, it is model-complete.
So suppose we have $f\colon A\to B$ and $g\colon A\to C$ as in the criterion. In the language of Lemma 11, we take $K_1 = B$, $K_2 = C$, $M_1 = f(A)$, and $M_2 = g(A)$. Then $M_1$ and $M_2$ are isomorphic (and the isomorphism respects the $S$ predicates, since they're included in $L$), so Lemma 11 tells us that we can extend this isomorphism to an $L$-isomorphism between the algebraic closures of $M_1$ and $M_2$ in $K_1$ and $K_2$. In other words, we can replace $A$ in our diagram by an extension $K_3$ whose underlying field is algebraically closed.
Now in particular, we have an embeddings of difference fields $K_3\to K_1$ and $K_3\to K_2$, and $K_3$ is algebraically closed. By Lemma 8, we can embed $K_1$ and $K_2$ in some difference field $K_4$, in such a way that the square over $K_3$ commutes. If $K_4$ is not e.c., we can further embed $K_4$ in an e.c. difference field $D$ (i.e. a model of $T$). To complete the proof, we just need to note that the embeddings $B\to D$ and $C\to D$ are $L$-embeddings, not just difference field embeddings. This is because $T$ is model-complete, so difference field embeddings between models of $T$ are elementary embeddings, so they preserve and reflect the definable predicates $S_{f,\overline{g},\overline{h}}$.