intros E F f n; try assumption.
case n.
intros H' H'0; try assumption.
case (not_injective_prop H'0).
intros x H'1; try assumption.
absurd (in_part x (full E)); auto with *.
apply not_in_part_comp_r with (empty E); auto with *.
inversion H'.
auto with *.
intros n0 H' H'0; try assumption.
case (not_injective_prop H'0).
intros x H'1; try assumption.
elim H'1; intros y E0; elim E0; intros H'2 H'3; try exact H'2; clear E0 H'1.
cut (cardinal (diff (full E) (single x)) n0).
intros H'1; try assumption.
case (cardinal_image_lesser f H'1).
intros m H'4; try assumption.
exists m; split; [ try assumption | idtac ].
elim H'4; intros H'5 H'6; try exact H'5; clear H'4.
apply cardinal_comp with (image f (diff (full E) (single x))) m; auto with *.
apply included_antisym.
unfold image_map in |- *.
apply image_included; auto with *.
unfold image_map in |- *.
unfold included in |- *.
intros x0 H'4; try assumption.
elim H'4.
intros x1 H'7; try assumption.
case (classic (Equal x1 x)).
intros H'8; try assumption.
simpl in |- *.
exists y; split; [ idtac | try assumption ].
split; [ idtac | try assumption ].
auto with *.
auto with *.
apply Trans with (f x); auto with *.
elim H'7; intros H'9 H'10; try exact H'10; clear H'7.
apply Trans with (f x1); auto with *.
intros H'8; try assumption.
simpl in |- *.
exists x1; split; [ idtac | try assumption ].
auto with *.
elim H'7; intros H'9 H'10; try exact H'10; clear H'7.
elim H'4; intros H'5 H'6; try exact H'6; clear H'4; auto with *.
apply cardinal_S with (full E) x; auto with *.
unfold add_part in |- *.
auto with *.
Qed.

Lemma cardinal_image_equal_injective :
forall (E F : Setoid) (f : MAP E F) (n : nat),
cardinal (full E) n -> cardinal (image_map f) n -> injective f.