unfold field_div in |- *.
intros x y H'; try assumption.
apply Trans with (ring_mult x (ring_mult (field_inverse y) y):K);
auto with algebra.
apply Trans with (ring_mult x (ring_unit K):K); auto with algebra.
Qed.
Hint Resolve FIELD_simpl_r: algebra.

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