Section Trois.
Variable H K : Group U.
Variable subH : subgroup U H Gr.
Variable subK : subgroup U K Gr.
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.
End Trois.
Section Quatre.
Variable H K : Group U.
Variable subH : subgroup U H Gr.
Variable subK : subgroup U K Gr.

(* 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).