Quantifier elimination for o-minimal structures expanded by a valuational cut

Let R be an o-minimal expansion of a group in language which Th(R) eliminates quantifiers, and let C predicate for valuational cut R. We identify condition that implies quantifier elimination Th(R,C) the expanded by small number constants, which, turn, is implied having being universally axiomatizable. The applies example case when convex subring field its residue o-minimal.