Section Lemmas1.
Variable K : FIELD.
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).
exact (field_inverse_r_prf K).
Qed.

Lemma FIELD_inverse_l :
forall x : K,
~ Equal x (monoid_unit K) ->
Equal (ring_mult (field_inverse x) x) (ring_unit K).