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

Lemma nat_to_group_add :
forall n m : nat,
Equal (nat_to_group (n + m))
(sgroup_law G (nat_to_group n) (nat_to_group m)).