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).

(* Specifications: *)
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.
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.
Lemma included2_union : forall A B C : part_set E, included A C -> included B C -> included (union A B) C.
Lemma union_comp : forall A A' B B' : part_set E, Equal A A' -> Equal B B' -> Equal (union A B) (union A' B').
Lemma union_assoc : forall A B C : part_set E, Equal (union A (union B C)) (union (union A B) C).
Lemma union_com : forall A B : part_set E, Equal (union A B) (union B A).