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