Section First_inductive_lemmas.
Variable U : Type.
Variable B : Ensemble U.
Variable D : PO U.

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.