Definition field_inverse (x : K) : K := Ap (field_inverse_map K) x.Definition field_div (x y : K) : K := ring_mult x (field_inverse y).

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)).