Section Lemmas.
Variable R : RING.
exact (sgroup_assoc_prf (E:=R) (ring_monoid R)).
Qed.
unfold ring_mult in |- *; auto with algebra.
Qed.
exact (monoid_unit_r_prf (ring_monoid R)).
Qed.
exact (monoid_unit_l_prf (ring_monoid R)).
Qed.
exact (ring_dist_r_prf R).
Qed.

Lemma RING_dist_l :
forall x y z : R,
Equal (ring_mult x (sgroup_law R y z))
(sgroup_law R (ring_mult x y) (ring_mult x z)).