Section First_inductive_lemma.
Variable U : Type.
Variable C : Cpo U.
Let D : PO U := PO_of_cpo U C.

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).