Section Group.
Variable G : ABELIAN_GROUP.
change
(forall x y : G:ABELIAN_SGROUP, Equal (sgroup_law _ x y) (sgroup_law _ y x))
in |- *; auto with algebra.
Qed.

Lemma ABELIAN_GROUP_permute :
forall x y z : G,
Equal (sgroup_law _ x (sgroup_law _ y z))
(sgroup_law _ y (sgroup_law _ x z)).