simpl in |- *; auto with algebra.
red in |- *.
simpl in |- *; auto with algebra.
intros x; try assumption.
elim x.
simpl in |- *; auto with algebra.
intros subtype_elt' subtype_prf'; red in |- *.
simpl in |- *; auto with algebra.
Qed.
Comments "The inclusion of parts:".
Definition included (A B : part_set E) : Prop :=
forall x : E, in_part x A -> in_part x B.
Comments "The relation of inclusion is an order relation:".

Lemma included_refl : forall A : part_set E, included A A.
simpl in |- *; unfold included in |- *; auto with algebra.