  By a theorem of Chevalley the image of a morphism of varieties is a
constructible set. The algebraic version of this fact is usually stated as a
result on "extension of specializations" or "lifting of prime ideals". We
present a difference analog of this theorem. The approach is based on the
philosophy that occasionally one needs to pass to higher powers of $\sigma$,
where $\sigma$ is the endomorphism defining the difference structure. In other
words, we consider difference pseudo fields (which are finite direct products
of fields) rather than difference fields. We also prove a result on
compatibility of pseudo fields and present some applications of the main
theorem, e.g. constrained extension and uniqueness of differential
Picard-Vessiot rings with a difference parameter.
