Section Second_inductive_lemma.
Variable U : Type.
Variable C : Cpo U.
Definition D : PO U := PO_of_cpo U C.

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).