intros.
simpl in |- *.
case (M b).
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_Map_to_equal_subsets: algebra.

Lemma Map_embed_eats_Map_to_equal_subsets :
forall (A B : Setoid) (X Y : part_set A) (H : X =' Y in _) (M : Map B X),
Map_embed (Map_to_equal_subsets H M) =' Map_embed M in _.