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

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