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

Lemma empty_inter :
forall A : part_set E, Equal (inter (empty E) A) (empty E).