simple induction m; simpl in |- *.
rewrite mult_comm; simpl in |- *; auto with algebra.
intros n0 H'; try assumption.
rewrite mult_comm; simpl in |- *; auto with algebra.
simpl in |- *.
apply Trans with (sgroup_law G (nat_to_group r n) (nat_to_group r (n0 * n)));
auto with algebra.
apply Trans with (sgroup_law G (nat_to_group r (n0 * n)) (nat_to_group r n));
auto with algebra.
apply SGROUP_comp; auto with algebra.
rewrite mult_comm; simpl in |- *; auto with algebra.
Qed.
Hint Resolve nat_to_group_mult: algebra.

(* Specifications: *)
Lemma nat_to_group_unit :
forall n : nat, Equal (nat_to_group (monoid_unit G) n) (monoid_unit G).