(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
Section Lemmas_on_partial_orders.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.

(* Specifications: *)
Theorem Rel_of_reflexive : forall x : U, Rel_of U D x x.
Hint Resolve Rel_of_reflexive.
Theorem Rel_of_antisymmetric : Antisymmetric U (Rel_of U D).
Hint Resolve Rel_of_antisymmetric.
Theorem Rel_of_transitive : Transitive U (Rel_of U D).
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).