Section First_inductive_lemma.
Variable U : Type.
Variable C : Cpo U.
Let D : PO U := PO_of_cpo U C.
Inductive Lubs_of_finite_parts (X : Ensemble U) : Ensemble U :=
Lubs_of_finite_parts_def :
forall (bsup : U) (Y : Ensemble U),
Included U Y X ->
Finite U Y -> Lub U D Y bsup -> In U (Lubs_of_finite_parts X) bsup.

Theorem LFP_directed :
forall X : Ensemble U,
(forall Y : Ensemble U,
Included U Y X -> Finite U Y -> exists bsup : U, Lub U D Y bsup) ->
Included U X (Carrier_of U D) ->
Consistent U D X -> Directed U D (Lubs_of_finite_parts X).