Definition included (A B : part_set E) : Prop :=
  forall x : E, in_part x A -> in_part x B.

(* Specifications: *)
Lemma included_refl : forall A : part_set E, included A A.
Lemma included_antisym : forall A B : part_set E, included A B -> included B A -> Equal A B.
Lemma included_trans : forall A B C : part_set E, included A B -> included B C -> included A C.
Lemma included_comp : forall A A' B B' : part_set E, Equal A A' -> Equal B B' -> included A B -> included A' B'.
Lemma eq_part_included : forall A B : part_set E, Equal A B -> included A B.
Lemma empty_included : forall A : part_set E, included (empty E) A.
Lemma full_included : forall A : part_set E, included A (full E).