intros x y z H' H'0; try assumption.
cut
(Equal (ring_mult z (sgroup_law R x (group_inverse R y))) (monoid_unit R)).
intros H'1; try assumption.
cut (Equal (sgroup_law R x (group_inverse R y)) (monoid_unit R)).
intros H'2; try assumption.
apply GROUP_reg_right with (group_inverse R y); auto with algebra.
apply Trans with (monoid_unit R); auto with algebra.
apply INTEGRAL_DOMAIN_mult_n0_l with z; auto with algebra.
apply
Trans with (sgroup_law R (ring_mult z x) (ring_mult z (group_inverse R y)));
auto with algebra.
apply
Trans with (sgroup_law R (ring_mult z x) (group_inverse R (ring_mult z y)));
auto with algebra.
apply
Trans with (sgroup_law R (ring_mult z x) (group_inverse R (ring_mult z x)));
auto with algebra.
Qed.

Lemma INTEGRAL_DOMAIN_mult_eq_r :
forall x y z : R,
Equal (ring_mult x z) (ring_mult y z) ->
Equal x y \/ Equal z (monoid_unit R).