Definition Reg_law (A : U -> Prop) := forall x y : U, x =_S y -> A x -> A y.Structure > Setoid_pred : Type := 
  {Pred :> U -> Prop; Prf_reg :> Reg_law Pred}.Structure > Setoid_pred : Type := 
  {Pred :> U -> Prop; Prf_reg :> Reg_law Pred}.Structure > Setoid_pred : Type := 
  {Pred :> U -> Prop; Prf_reg :> Reg_law Pred}.Structure SubType : Type :=  {Elt_sub : U; Prf_constr : A Elt_sub}.Structure SubType : Type :=  {Elt_sub : U; Prf_constr : A Elt_sub}.Structure SubType : Type :=  {Elt_sub : U; Prf_constr : A Elt_sub}.Definition Equal_SubType (a b : SubType) := Elt_sub a =_S Elt_sub b.

(* Specifications: *)
Lemma Equal_SubType_equiv : Equivalence Equal_SubType.