Inductive Power_set (A : Ensemble U) : Ensemble (Ensemble U) :=
       Definition_of_Power_set :
         forall X : Ensemble U,
         Included U X A -> In (Ensemble U) (Power_set A) X.Inductive Power_set (A : Ensemble U) : Ensemble (Ensemble U) :=
       Definition_of_Power_set :
         forall X : Ensemble U,
         Included U X A -> In (Ensemble U) (Power_set A) X.

Theorem Union_is_Lub : forall a b : Ensemble U, Included U a A -> Included U b A -> Lub (Ensemble U) Power_set_PO (Couple (Ensemble U) a b) (Union U a b).