intros.
elim x.
simpl in |- *.
auto with algebra.
Qed.
Hint Resolve subtype_elt_eats_map_between_equal_subsets: algebra.

Lemma map_between_equal_subsets_inj :
forall (A : Setoid) (X Y : part_set A) (H H' : X =' Y in _) (x x' : X),
map_between_equal_subsets H x =' map_between_equal_subsets H' x' in _ ->
x =' x' in _.