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.
exact (field_inverse_l_prf K).
Qed.
exact (field_unit_non_zero K).
Qed.
unfold field_inverse in |- *.
auto with algebra.
Qed.
Hint Resolve FIELD_comp FIELD_inverse_r FIELD_inverse_l FIELD_unit_non_zero:
algebra.
apply Trans with (ring_mult (field_inverse (ring_unit K)) (ring_unit K));
auto with algebra.
Qed.
Hint Resolve FIELD_unit_inverse: algebra.

Lemma FIELD_reg_left :
forall x y z : K,
~ Equal x (monoid_unit K) ->
Equal (ring_mult x y) (ring_mult x z) -> Equal y z.