Definition D : PO U := PO_of_cpo U C.

(* Specifications: *)
Theorem Triples_are_enough_finite_case : (forall x y z : U, Consistent U D (Triple U x y z) -> ex (fun bsup : U => Lub U D (Triple U x y z) bsup)) -> forall X : Ensemble U, Finite U X -> Included U X (Carrier_of U D) -> Consistent U D X -> ex (fun bsup : U => Lub U D X bsup).
Theorem Triples_are_enough : (forall x y z : U, Consistent U D (Triple U x y z) -> ex (fun bsup : U => Lub U D (Triple U x y z) bsup)) -> Coherent U D.