unfold field_div in |- *; auto with algebra.
Qed.
Hint Resolve FIELD_one_div_x: algebra.

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