Inductive powers (a : U) : Ensemble U :=
    In_powers : forall (m : Z) (x : U), x = exp m a -> In U (powers a) x.Inductive powers (a : U) : Ensemble U :=
    In_powers : forall (m : Z) (x : U), x = exp m a -> In U (powers a) x.

(* Specifications: *)
Theorem powers_of_one_element : forall a : U, In U G a -> Setsubgroup U (powers a) Gr.
Theorem T_1_6_3 : Setsubgroup U H Gr.
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.