intros r r' n H'; try assumption.
elim n.
apply Trans with (monoid_unit G); auto with algebra.
intros p; try assumption.
apply Trans with (nat_to_group r (nat_of_P (pos_abs (ax3 p))));
auto with algebra.
apply Trans with (nat_to_group r' (nat_of_P (pos_abs (ax3 p))));
auto with algebra.
intros p; try assumption.
apply
Trans with (group_inverse G (nat_to_group r (nat_of_P (pos_abs (ax3 p)))));
auto with algebra.
apply
Trans with (group_inverse G (nat_to_group r' (nat_of_P (pos_abs (ax3 p)))));
auto with algebra.
Qed.
Hint Resolve Z_to_group_nat_fun_comp: algebra.

Lemma Z_to_group_nat_neg :
forall (p : positive) (r : G),
Equal (Z_to_group_nat_fun r (Zneg p))
(Z_to_group_nat_fun (group_inverse G r) (Zpos p)).