  Let R be an o-minimal field with a proper convex subring V. We axiomatize the
class of all structures (R,V) such that k_ind, the corresponding residue field
with structure induced from R via the residue map, is o-minimal. More
precisely, in previous work it was shown that certain first order conditions on
(R,V) are sufficient for the o-minimality of k_ind. Here we prove that these
conditions are also necessary.
