Section Lemmas_on_partial_orders.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_reflexive.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_antisymmetric.
elim D; simpl in |- *; auto.
intros C R H' H'0; elim H'0; auto 10.
Qed.
Hint Resolve Rel_of_transitive.

Theorem Couple_included_in_carrier :
forall x y : U,
In U (Carrier_of U D) x ->
In U (Carrier_of U D) y -> Included U (Couple U x y) (Carrier_of U D).