proposition: Lemma FIELD_div_div2 : forall x y z : K, ~ Equal y (monoid_unit K) -> ~ Equal z (monoid_unit K) -> Equal (field_div (field_div x y) z) (field_div x (ring_mult z y)).
proof: Proof. unfold field_div in |- *. intros x y z H' H'0; try assumption. apply Trans with (ring_mult x (ring_mult (field_inverse y) (field_inverse z)):K); auto with algebra. apply Sym; auto with algebra. Qed.
imports: Field_cat Ring_facts