Inductive Prod : Ensemble U :=
    Definition_of_Prod :
      forall x y z : U,
      In U (G_ U H) x -> In U (G_ U K) y -> star x y = z -> In U Prod z.Inductive Prod : Ensemble U :=
    Definition_of_Prod :
      forall x y z : U,
      In U (G_ U H) x -> In U (G_ U K) y -> star x y = z -> In U Prod z.

(* Specifications: *)
Theorem T4 : Same_set U (Prod H K) (Prod K H) -> Setsubgroup U (Prod H K) Gr.
Theorem T4R : Setsubgroup U (Prod H K) Gr -> Included U (Prod H K) (Prod K H).
Theorem T4R1 : Setsubgroup U (Prod H K) Gr -> Included U (Prod K H) (Prod H K).
Theorem T_1_6_8 : Same_set U (Prod H K) (Prod K H) <-> Setsubgroup U (Prod H K) Gr.