simple induction n.
intros.
apply cardinal_empty.
inversion H1.
elim (classic (Equal x0 x)); intros.
apply Trans with B0.
apply add_part_simpl with x; auto with algebra.
apply not_in_part_trans_eq with x0; auto with algebra.
apply Trans with A; auto with algebra.
apply Trans with (add_part B0 x0); auto with algebra.
auto with algebra.
absurd (in_part x (single x0)).
intuition.
apply in_part_comp_r with (add_part B0 x0); auto with algebra.
apply in_part_comp_r with A; auto with algebra.
apply in_part_comp_r with (add_part B x); auto with algebra.
apply Trans with (add_part (empty E) x0); auto with algebra.
intros.
inversion H2.
elim (classic (Equal x x0)); intros.
apply cardinal_comp_l with B0; auto with algebra.
apply add_part_simpl with x; auto with algebra.
apply not_in_part_trans_eq with x0; auto with algebra.
apply Trans with A; auto with algebra.
apply Trans with (add_part B0 x0); auto with algebra.
elim (cardinal_sup3 (A:=A) (B:=B) (C:=B0) (x:=x) (y:=x0)); auto with algebra.
intros C H9.
elim H9; clear H9; intros.
elim H10; clear H10; intros.
elim H11; clear H11; intros.
apply cardinal_add with C x0; auto with algebra.
apply H with B0 x; auto with algebra.
Qed.

Lemma cardinalO_unique :
forall A : part_set E, cardinal A 0 -> forall m : nat, cardinal A m -> 0 = m.