simple induction n; simpl in |- *; auto with *.
intros n0 H' r0; try assumption.
apply
Trans
with
(sgroup_law G (group_inverse G r0) (group_inverse G (nat_to_group r0 n0)));
auto with *.
apply
Trans
with
(sgroup_law G (group_inverse G r0) (nat_to_group (group_inverse G r0) n0));
auto with *.
Qed.
Hint Resolve nat_to_group_inverse: algebra.

Lemma Z_to_group_fun_eq :
forall z : ZZ, Equal (Z_to_group_fun z) (Z_to_group_nat r z).