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