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_Put_behind_mono : forall (m:Map A) (m':Map B) (a:ad) (y:A) (y':B), MapSubset A B m m' -> MapSubset A B (MapPut_behind A m a y) (MapPut_behind B m' a y').