simpl in |- *; intuition.
Qed.

(* Specifications: *)
Lemma in_part_union_r :
forall (A B : part_set E) (x : E), in_part x B -> in_part x (union A B).
Lemma in_part_inter_r :
forall (A B : part_set E) (x : E), in_part x (inter A B) -> in_part x B.
Lemma in_part_inter :
forall (A B : part_set E) (x : E),
in_part x A -> in_part x B -> in_part x (inter A B).
Lemma inter_not_in_r :
forall (A B : part_set E) (x : E), ~ in_part x B -> ~ in_part x (inter A B).
Lemma included2_inter :
forall A B C : part_set E,
included A C -> included B C -> included (inter A B) C.