unfold disjoint in |- *.
intros A A' B B' H' H'0 H'1; try assumption.
apply Trans with (inter A B).
auto with *.
auto with *.
Qed.

(* Specifications: *)
Lemma empty_not_in :
forall A : part_set E, Equal A (empty E) -> forall x : E, ~ in_part x A.