Section Second.
Variable H : Ensemble U.
Variable witness : U.
Variable h0 : In U H witness.
Variable h1 : Included U H G.
Variable h2 : forall a b : U, In U H a -> In U H b -> In U H (star a (inv b)).
Let eH : U := star witness (inv witness).

(* Specifications: *)
Theorem T_1_6_3 : Setsubgroup U H Gr.
End Second.
Theorem Ex1 : Setsubgroup U (Singleton U e) Gr.
Theorem Ex2 : Setsubgroup U (Singleton U e) Gr.
Lemma Ex3 : forall n : Z, exp n e = e.
Lemma Ex4 : powers e = Singleton U e.
Theorem Ex5 : Setsubgroup U (Singleton U e) Gr.