unfold eq_part in |- *; auto with algebra.
simpl in |- *; unfold not in |- *.
intros A x; try assumption.
split; [ try assumption | idtac ].
apply NNPP.
tauto.
Qed.
Hint Resolve compl_compl: algebra.

(* Specifications: *)
Lemma compl_not_in :
forall (A : part_set E) (x : E), in_part x A -> ~ in_part x (compl A).