unfold field_div in |- *.
intros x y H' H'0; try assumption.
apply Trans with (field_inverse (ring_mult x y):K); auto with algebra.
apply Trans with (ring_mult (field_inverse y) (field_inverse x):K);
auto with algebra.
Qed.
Hint Resolve FIELD_one_div_xy: algebra.

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