intros x y z t; try assumption.
apply Trans with (sgroup_law S x (sgroup_law S y (sgroup_law S z t)));
auto with algebra.
apply Trans with (sgroup_law S x (sgroup_law S (sgroup_law S y z) t));
auto with algebra.
apply Trans with (sgroup_law S x (sgroup_law S (sgroup_law S z y) t));
auto with algebra.
apply SGROUP_comp; auto with algebra.
apply SGROUP_comp; auto with algebra.
apply ABELIAN_SGROUP_com.
apply Trans with (sgroup_law S x (sgroup_law S z (sgroup_law S y t)));
auto with algebra.
Qed.
End Sgroup.
Hint Immediate ABELIAN_SGROUP_com ABELIAN_SGROUP_permute ABELIAN_SGROUP4:
algebra.
Section Monoid.
Variable M : ABELIAN_MONOID.

Lemma ABELIAN_MONOID_com :
forall x y : M, Equal (sgroup_law _ x y) (sgroup_law _ y x).