  We prove a direct image theorem stating that the direct image of a Galois
formula by a morphism of difference schemes is equivalent to a Galois formula
over fields with powers of Frobenius. As a consequence, we obtain an effective
quantifier elimination procedure and a precise algebraic-geometric description
of definable sets over fields with Frobenii in terms of twisted Galois formulae
associated with finite Galois covers of difference schemes.
