Definition eqmap (m m':Map A) := eqm A (MapGet A m) (MapGet A m').

Lemma MapDomRestrTo_By : forall (m:Map A) (m':Map B) (m'':Map C), eqmap (MapDomRestrTo A C (MapDomRestrBy A B m m') m'') (MapDomRestrTo A C m (MapDomRestrBy C B m'' m')).