intros E F f A n H'; try assumption.
apply
cardinal_ind2
with
(P := fun (n : nat) (A : part_set E) (c : cardinal A n) =>
ex (fun m : nat => cardinal (image f A) m /\ m <= n)).
intros A0 H'0; try assumption.
exists 0; split; [ idtac | auto with * ].
apply cardinal_empty; auto with *.
intros n0 H'0 A0 B x H'1 H'2 H'3; try assumption.
case (classic (in_part (f x) (image f B))).
intros H'4; try assumption.
elim (H'0 B);
[ intros m E0; elim E0; intros H'7 H'8; try exact H'7; clear E0 | idtac ].
exists m; split; [ idtac | try assumption ].
apply cardinal_comp with (image f B) m; auto with *.
apply Sym.
apply Trans with (image f (add_part B x)); auto with *.
unfold add_part in |- *.
apply Trans with (union (image f B) (image f (single x))); auto with *.
apply Trans with (union (image f B) (single (Ap f x))); auto with *.
auto with *.
apply cardinal_S with A0 x; auto with *.
intros H'4; try assumption.
elim (H'0 B);
[ intros m E0; elim E0; intros H'7 H'8; try exact H'7; clear E0 | idtac ].
exists (S m); split; [ try assumption | idtac ].
apply cardinal_add with (image f B) (Ap f x); auto with *.
unfold add_part in |- *.
unfold add_part in H'2.
apply Trans with (image f (union B (single x))); auto with *.
apply Trans with (union (image f B) (image f (single x))); auto with *.
auto with *.
apply cardinal_S with A0 x; auto with *.
auto with *.
Qed.

Lemma cardinal_image_injective :
forall (E F : Setoid) (f : MAP E F) (A : part_set E) (n : nat),
cardinal A n -> injective f -> cardinal (image f A) n.