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

Lemma ax1 : ~ (0 > 0)%Z.
red in |- *.