simple induction n; simpl in |- *.
intros r; try assumption.
apply Trans with (monoid_unit G); auto with algebra.
auto with algebra.
intros p r; try assumption.
apply
Trans
with (Z_to_group_nat_fun (group_inverse G (group_inverse G r)) (Zpos p));
auto with algebra.
Qed.
Hint Resolve Z_to_group_nat_inv: algebra.

Lemma nat_to_group_mult :
forall (r : G) (n m : nat),
Equal (nat_to_group r (n * m)) (nat_to_group (nat_to_group r n) m).