intros.
inversion H.
exists x.
apply Trans with (add_part B x); auto with algebra.
apply Trans with (add_part (empty E) x); auto with algebra.
Qed.
(* Some lemmas. *)

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