Definition MapSubset (m:Map A) (m':Map B) :=
    forall a:ad, in_dom A a m = true -> in_dom B a m' = true.Definition MapSubset_1 (m:Map A) (m':Map B) :=
    match MapSweep A (fun (a:ad) (_:A) => negb (in_dom B a m')) m with
    | None => true
    | _ => false
    end.Definition MapSubset_2 (m:Map A) (m':Map B) :=
    eqmap A (MapDomRestrBy A B m m') (M0 A).

Lemma MapSubset_DomRestrBy_mono : forall (m:Map A) (m':Map B) (m'':Map C) (m''':Map D), MapSubset _ _ m m'' -> MapSubset _ _ m''' m' -> MapSubset _ _ (MapDomRestrBy _ _ m m') (MapDomRestrBy _ _ m'' m''').