Definition Ensemble := U -> Prop.Definition In (B : Ensemble) (x : U) : Prop := B x.Definition Included (B C : Ensemble) : Prop :=
     forall x : U, In B x -> In C x.Inductive Empty_set : Ensemble :=.Inductive Singleton (x : U) : Ensemble :=
       In_singleton : In (Singleton x) x.Inductive Singleton (x : U) : Ensemble :=
       In_singleton : In (Singleton x) x.Inductive Union (B C : Ensemble) : Ensemble :=
     | Union_introl : forall x : U, In B x -> In (Union B C) x
     | Union_intror : forall x : U, In C x -> In (Union B C) x.Inductive Union (B C : Ensemble) : Ensemble :=
     | Union_introl : forall x : U, In B x -> In (Union B C) x
     | Union_intror : forall x : U, In C x -> In (Union B C) x.Inductive Union (B C : Ensemble) : Ensemble :=
     | Union_introl : forall x : U, In B x -> In (Union B C) x
     | Union_intror : forall x : U, In C x -> In (Union B C) x.Inductive Intersection (B C : Ensemble) : Ensemble :=
       Intersection_intro :
         forall x : U, In B x -> In C x -> In (Intersection B C) x.Inductive Intersection (B C : Ensemble) : Ensemble :=
       Intersection_intro :
         forall x : U, In B x -> In C x -> In (Intersection B C) x.Inductive Couple (x y : U) : Ensemble :=
     | Couple_l : In (Couple x y) x
     | Couple_r : In (Couple x y) y.Inductive Couple (x y : U) : Ensemble :=
     | Couple_l : In (Couple x y) x
     | Couple_r : In (Couple x y) y.Inductive Couple (x y : U) : Ensemble :=
     | Couple_l : In (Couple x y) x
     | Couple_r : In (Couple x y) y.Inductive Setminus (B C : Ensemble) : Ensemble :=
       Setminus_intro :
         forall x : U, In B x -> ~ In C x -> In (Setminus B C) x.Inductive Setminus (B C : Ensemble) : Ensemble :=
       Setminus_intro :
         forall x : U, In B x -> ~ In C x -> In (Setminus B C) x.Inductive Disjoint (B C : Ensemble) : Prop :=
       Disjoint_intro :
         (forall x : U, ~ In (Intersection B C) x) -> Disjoint B C.Inductive Disjoint (B C : Ensemble) : Prop :=
       Disjoint_intro :
         (forall x : U, ~ In (Intersection B C) x) -> Disjoint B C.Inductive Non_empty (B : Ensemble) : Prop :=
       Non_empty_intro : forall x : U, In B x -> Non_empty B.Inductive Non_empty (B : Ensemble) : Prop :=
       Non_empty_intro : forall x : U, In B x -> Non_empty B.Inductive Finite : Ensemble -> Prop :=
     | Empty_is_finite : Finite Empty_set
     | Union_is_finite :
         forall A : Ensemble,
         Finite A -> forall x : U, ~ In A x -> Finite (Union (Singleton x) A).Inductive Finite : Ensemble -> Prop :=
     | Empty_is_finite : Finite Empty_set
     | Union_is_finite :
         forall A : Ensemble,
         Finite A -> forall x : U, ~ In A x -> Finite (Union (Singleton x) A).Inductive Finite : Ensemble -> Prop :=
     | Empty_is_finite : Finite Empty_set
     | Union_is_finite :
         forall A : Ensemble,
         Finite A -> forall x : U, ~ In A x -> Finite (Union (Singleton x) A).(*************************************************************************)
 (*** Initialement, la definition de Same_set etait :
   
   Inductive Same_set [B,C: Ensemble]: Prop :=
         Same_set_intro: (Included B C) -> (Included C B) -> (Same_set B C).

  *** mais dans le lemme Same_set_equivalence de ps.v (par exemple)
      	(Same_set U)
      n'est plus unifie avec
      	[x,y:(Ensemble U)](and (Included x y) (Included y x))

  *** on redefinit donc Same_set et Same_Set_intro de la maniere
      suivante : ***)
 (*************************************************************************)

   Definition Same_set (B C : Ensemble) : Prop :=
     Included B C /\ Included C B.

Lemma Same_set_intro : forall B C : Ensemble, Included B C -> Included C B -> Same_set B C.