intros A; try assumption.
apply in_eq_part.
intros x H'; try assumption.
apply in_part_inter_l with A; auto with *.
intros x H'; try assumption.
absurd (in_part x (empty E)); auto with *.
Qed.
Hint Resolve empty_inter: algebra.

(* Specifications: *)
Lemma in_part_trans_eq :
forall (A : part_set E) (x y : E), in_part x A -> Equal y x -> in_part y A.