Parameter
  in_part_union_or :
    forall (A B : part_set E) (x : E),
    in_part x A \/ in_part x B -> in_part x (union A B).

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