Theorem Pairs_are_enough_finite_case : Cpo U D -> (forall x y : U, Compatible U D x y -> exists bsup : U, Lub U D (Couple U x y) bsup) -> forall X : Ensemble U, Finite U X -> Included U X (Carrier_of U D) -> (exists maj : U, Upper_Bound U D X maj) -> exists bsup : U, Lub U D X bsup.