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_inv_div : forall x y : K, ~ Equal x (monoid_unit K) -> ~ Equal y (monoid_unit K) -> Equal (field_inverse (field_div x y)) (field_div y x).