Section Inclusion.
Variable E : Setoid.
Comments "The relation of belonging is compatible with equality:".
intros A; try assumption.
exact (Pred_compatible_prf (E:=E) (p:=A)).
Qed.
simpl in |- *; unfold eq_part in |- *.
intros x A B H' H'0; try assumption.
elim (H'0 x).
intuition.
Qed.
unfold not in |- *; auto with algebra.
Qed.
Hint Resolve empty_prop: algebra.

Lemma full_prop : forall x : E, in_part x (full E).