intros E F f A n H' H'0; try assumption.
case (cardinal_image_lesser f H').
intros x H'1; try assumption.
elim H'1; intros H'2 H'3; try exact H'2; clear H'1.
cut (x < n \/ x = n).
intros H'1; try assumption.
elim H'1; [ intros H'4; try exact H'4; clear H'1 | intros H'4; clear H'1 ].
case
(tiroirs (E:=E) (F:=F) (f:=f) (n:=n) (Chaussettes:=A) H' (m:=x)
(Tiroirs:=image f A) H'2 H'4).
auto with *.
intros x0 H'1; try assumption.
elim H'1; intros y E0; elim E0; intros H'5 H'6; try exact H'5; clear E0 H'1.
red in H'0.
absurd (Equal x0 y); auto with *.
apply cardinal_comp with (image f A) x; auto with *.
case (lt_eq_lt_dec x n).
intros H'1; elim H'1;
[ intros H'4; try exact H'4; clear H'1 | intros H'4; clear H'1 ];
auto with *.
intros H'1; try assumption.
absurd (n < x); auto with *.
Qed.
Parameter
not_in_part_comp_r :
forall (E : Setoid) (A B : part_set E) (x : E),
~ in_part x A -> Equal A B -> ~ in_part x B.
Parameter
diff_single_not_in :
forall (E : Setoid) (A : part_set E) (x : E),
~ in_part x (diff A (single x)).
Hint Resolve diff_single_not_in: algebra.
Parameter
diff_el_union_single :
forall (E : Setoid) (A : part_set E) (x : E),
in_part x A -> Equal A (union (diff A (single x)) (single x)).
Hint Resolve diff_el_union_single: algebra.

Lemma cardinal_image_strict_lesser :
forall (E F : Setoid) (f : MAP E F) (n : nat),
cardinal (full E) n ->
~ injective f -> exists m : nat, cardinal (image_map f) m /\ m < n.