proposition: Lemma INTEGRAL_DOMAIN_mult_eq_l : forall x y z : R, Equal (ring_mult z x) (ring_mult z y) -> Equal x y \/ Equal z (monoid_unit R).
proof: Proof. intros x y z H'; try assumption. cut (Equal (ring_mult z (sgroup_law R x (group_inverse R y))) (monoid_unit R)). intros H'0; try assumption. elim (INTEGRAL_DOMAIN_prop (x:=z) (y:=sgroup_law R x (group_inverse R y))); auto with algebra. intros H'1; try assumption. left; try assumption. apply GROUP_reg_right with (group_inverse R y); auto with algebra. apply Trans with (monoid_unit R); 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.
imports: Integral_domain_cat Ring_facts Classical_Prop