intros n g; try assumption.
apply Trans with (Ap (sgroup_map (monoid_sgroup_hom (Z_to_group_nat g))) n);
auto with algebra.
apply Sym.
apply Z_to_group_fun_eq.
Qed.
Hint Resolve Z_to_group_nat_eq_pos: algebra.

Lemma Zopp1 : forall n : Z, (n < 0)%Z -> (- n > 0)%Z.
simple induction n.