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

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