  We use the method of characteristic sets with respect to two term orderings
to prove the existence and obtain a method of computation of a bivariate
Kolchin-type dimension polynomial associated with a non-reflexive
difference-differential ideal in the algebra of difference-differential
polynomials with several basic derivations and one translation. In particular,
we obtain a new proof and a method of computation of the dimension polynomial
of a non-reflexive prime difference ideal in the algebra of difference
polynomials over an ordinary difference field. As a consequence, it is shown
that the reflexive closure of a prime difference polynomial ideal is the
inverse image of this ideal under a power of the basic translation. We also
discuss applications of our results to the analysis of systems of algebraic
difference-differential equations.
