On the for all there exists-Theories of Free Projective Planes

Studying the elementary properties of free projective planes of finite rank, we prove that for m > n, an arbitrary for all there exists for all-formula phi(& x233;) and a tuple u of elements of the free projective plane Fn if phi(u) holds on the plane Fm then phi(u) holds on the plane Fn too. This implies the coincidence of the for all there exists-theories of free projective planes of different finite ranks.