Definition union : part_set E -> part_set E -> part_set E.

(* Specifications: *)
Lemma included_union_l : forall A B : part_set E, included A (union A B).
Lemma included_union_r : forall A B : part_set E, included B (union A B).
Lemma in_part_union_l : forall (A B : part_set E) (x : E), in_part x A -> in_part x (union A B).
Lemma in_part_union_r : forall (A B : part_set E) (x : E), in_part x B -> in_part x (union A B).