intros A x; red in |- *.
unfold add_part in |- *.
auto with *.
Qed.
Hint Resolve included_add_part: algebra.

(* Specifications: *)
Lemma union_not_in :
forall (A B : part_set E) (x : E),
~ in_part x A -> ~ in_part x B -> ~ in_part x (union A B).