We let R be an o-minimal expansion of a field, V a convex subring, and (R 0,V 0) an elementary substructure of (R,V). Our main result is that (R,V) considered as a structure in a language containing constants for all elements of R 0 is model complete relative to quantifier elimination in R, provided that k R (the residue field with structure induced from R) is o-minimal. Along the way we show that o-minimality of k R implies that the sets definable in k R are the same as the sets definable in k

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... with structure induced from (R,V). We also give a criterion for a superstructure of (R,V) being an elementary extension of (R,V).