simpl in |- *.
intros p q g; try assumption.
apply
Trans
with
(nat_to_group g
(nat_of_P
(pos_abs
(ax3
((fun (x : positive) (_ : positive -> positive)
(y : positive) => (x * y)%positive) p
(fun y : positive => y) q)))));
auto with algebra.
apply
Trans
with
(nat_to_group (Z_to_group_nat_fun g (Zpos p))
(nat_of_P (pos_abs (ax3 q)))); auto with algebra.
simpl in |- *.
rewrite
(fun (x y : positive) (_ : positive -> positive) =>
nat_of_P_mult_morphism x y).
apply Trans with (nat_to_group (nat_to_group g (nat_of_P p)) (nat_of_P q));
auto with algebra.
Qed.
Hint Resolve Z_group_nat_fun_mult_pos: algebra.

Lemma group_power_mult :
forall (g : G) (n m : ZZ),
Equal (group_power G g (ring_mult n m))
(group_power G (group_power G g n) m).