intros.
simpl in |- *.
red in |- *.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve Map_embed_cast_map_to_subset_inv: algebra.

Lemma Map_embed_eats_cast_map_to_subset :
forall (A D : Setoid) (B C : part_set A) (v : MAP D B)
(H : forall i : D, in_part (Map_embed v i) C),
Map_embed (cast_map_to_subset H) =' Map_embed v in _.