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

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