intros.
inversion H0; auto with algebra.
absurd (in_part x (empty E)); 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.
Qed.

Lemma cardinal_unique :
forall (n : nat) (A : part_set E),
cardinal A n -> forall m : nat, cardinal A m -> n = m.