simpl in |- *; intuition.
Qed.
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 in_part_union :
forall (A B : part_set E) (x : E),
in_part x (union A B) -> in_part x A \/ in_part x B.