 The elementary amalgamation theorem:   Suppose L is a first-order language, A is an L-structure and B, C are two elementary extensions of A. Then there are an elementary extension D of B and an elementary embedding e of C into D such that (i) for each element a of A, e(a) = a, and (ii) if c is an element of C but not of A, then e(c) is not in B.