  Let R be an o-minimal field and V a proper convex subring with residue field
k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion of
k by the standard parts of the definable relations in R. We investigate the
definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of
k_{ind}. We also show that if R is omega-saturated and V is the convex hull of
the rationals in R, then the sets definable in k_{ind} are exactly the standard
parts of the sets definable in (R,V).
