intros p p0; try assumption.
case (Z_gt_le_dec (Zpos p) (Zpos p0)); intros z.
rewrite ax5; auto with *.
apply Trans with (nat_to_group (nat_of_P (pos_abs (ax3 (p - p0)))));
auto with *.
simpl in |- *.
rewrite nat_of_P_minus_morphism; auto with *.
apply nat_to_group_minus.
apply nat_of_P_gt_Gt_compare_morphism.
auto with *.
case (Z_le_lt_eq_dec _ _ z); intros.
rewrite ax6; auto with *.
apply
Trans
with (group_inverse G (nat_to_group (nat_of_P (pos_abs (ax3 (p0 - p))))));
auto with *.
simpl in |- *.
rewrite nat_of_P_minus_morphism; auto with *.
apply
Trans
with
(group_inverse G
(sgroup_law G (nat_to_group (nat_of_P p0))
(group_inverse G (nat_to_group (nat_of_P p)))));
auto with *.
apply GROUP_comp.
apply nat_to_group_minus.
apply nat_of_P_gt_Gt_compare_morphism.
auto with *.
apply
Trans
with
(sgroup_law G
(group_inverse G (group_inverse G (nat_to_group (nat_of_P p))))
(group_inverse G (nat_to_group (nat_of_P p0))));
auto with *.
injection e.
intros H'; try assumption.
rewrite <- H'.
rewrite ax7.
apply Trans with (monoid_unit G); auto with *.
Qed.
Hint Resolve Zl4: algebra.

Lemma nat_to_group_com3 :
forall n : nat,
Equal (sgroup_law G (nat_to_group n) (group_inverse G r))
(sgroup_law G (group_inverse G r) (nat_to_group n)).