  This paper is the 10th in a sequence on the structure of sets of solutions to
systems of equations over groups, projections of such sets (Diophantine sets),
and the structure of definable sets over few classes of groups. In the 10th
paper we answer affirmatively a question of R. L. Vaught on the elementary
equivalence of free products of pairs of elementarily equivalent groups, and
obtain a generalization of Tarski's problem on the elementary equivalence of
non-abelian free groups. Finally, we prove that free products of stable groups
is stable, generalizing a previous result on the stability of free groups.
