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

Lemma MapDomRestrBy_By : forall (m:Map A) (m' m'':Map B), eqmap (MapDomRestrBy A B (MapDomRestrBy A B m m') m'') (MapDomRestrBy A B m (MapMerge B m' m'')).