intros x y; try assumption.
apply Sym.
apply GROUP_law_inverse.
apply Trans with (ring_mult (sgroup_law R x (group_inverse R x)) y);
auto with algebra.
apply Trans with (ring_mult (monoid_unit R) y); auto with algebra.
Qed.
Hint Resolve RING_op_mult_l: algebra.

Lemma RING_op_mult_r :
forall x y : R,
Equal (ring_mult x (group_inverse R y)) (group_inverse R (ring_mult x y)).