unfold group_power in |- *.
intros g n; try assumption.
simpl in |- *.
apply Trans with (Z_to_group_nat_fun g (group_inverse ZZ n));
auto with algebra.
apply Trans with (Z_to_group_nat_fun (group_inverse G g) n);
auto with algebra.
Qed.
Hint Resolve group_power_inv: algebra.

(* Specifications: *)
Lemma Z_group_nat_fun_mult_pos :
forall (p q : positive) (g : G),
Equal (Z_to_group_nat_fun g (Zpos p * Zpos q)%Z)
(Z_to_group_nat_fun (Z_to_group_nat_fun g (Zpos p)) (Zpos q)).