Section tiroirs_def.
Variable E F : Setoid.
Variable f : MAP E F.
intros E0 A x H'; try assumption.
apply in_eq_part.
intros x0 H'0; try assumption.
elim (classic (Equal x x0)); intros.
apply in_part_trans_eq with x; auto with *.
cut (in_part x0 (diff A (single x))).
unfold add_part in |- *; auto with *.
apply in_diff; auto with *.
intros x0 H'0; try assumption.
elim (classic (Equal x x0)); intros.
apply in_part_trans_eq with x; auto with *.
cut (in_part x0 (diff A (single x))).
intros H'1; try assumption.
apply diff_in_l with (single x).
auto with *.
unfold add_part in H'0.
elim (in_part_union H'0); intros.
auto with *.
absurd (Equal x x0); auto with *.
Qed.
Hint Resolve diff_add_part2: algebra.
intros B x n H' H'0; try assumption.
apply cardinal_S with B x; auto with *.
unfold not in |- *; intros.
cut (~ in_part x (single x)).
unfold not in |- *; auto with *.
apply diff_in_r with B; auto with *.
Qed.
Hint Resolve cardinal_minus_part: algebra.

Lemma tiroirs :
forall (n : nat) (Chaussettes : part_set E),
cardinal Chaussettes n ->
forall (m : nat) (Tiroirs : part_set F),
cardinal Tiroirs m ->
m < n ->
(forall x : E, in_part x Chaussettes -> in_part (f x) Tiroirs) ->
exists x : E, (exists y : E, ~ Equal x y /\ Equal (f x) (f y)).