intros E A n H'; try assumption.
apply
cardinal_ind2
with
(P := fun (n : nat) (A : part_set E) (c : cardinal A n) =>
forall B : part_set E,
included B A -> ex (fun m : nat => cardinal B m /\ m <= n)).
intros A0 H'0 B H'1; exists 0; split; [ try assumption | idtac ]; auto with *.
apply cardinal_empty.
inversion H'0.
cut (included B (empty E)); auto with *.
apply included_comp with B A0; auto with *.
intros n0 H'0 A0 B x H'1 H'2 H'3 B0 H'4; try assumption.
case (classic (Equal B0 A0)); intros.
exists (S n0); split; [ idtac | try assumption ].
apply cardinal_comp with A0 (S n0); auto with *.
auto with *.
case (not_included_exist H'4 H).
intros x0 H'5; try assumption.
elim H'5; intros H'6 H'7; try exact H'6; clear H'5.
lapply (H'0 (minus_part A0 x0));
[ intros H'8; elim (H'8 B0);
[ intros m E0; elim E0; intros H'11 H'12; try exact H'11; clear E0
| idtac ]
| idtac ]; auto with *.
exists m; split; [ try assumption | idtac ].
apply le_trans with n0; auto with *.
unfold minus_part in |- *; simpl in |- *.
red in |- *.
intros x1 H'5; try assumption.
simpl in |- *.
split; [ try assumption | idtac ].
apply H'4; auto with *.
red in |- *.
intros H'9; try assumption.
apply H'7.
apply in_part_comp_l with x1; auto with *.
apply cardinal_S with A0 x0; auto with *.
apply Sym.
apply minus_add; auto with *.
auto with *.
Qed.

Lemma map_not_injective :
forall (A B : Setoid) (f : MAP A B) (n m : nat),
cardinal (full A) n -> cardinal (full B) m -> m < n -> ~ injective f.