intros x y H' H'0; try assumption.
apply FIELD_law_inverse.
apply
Trans
with
(ring_mult x
(ring_mult y (ring_mult (field_inverse y) (field_inverse x)))
:K); auto with algebra.
apply
Trans
with
(ring_mult x
(ring_mult (ring_mult y (field_inverse y)) (field_inverse x))
:K); auto with algebra.
apply Trans with (ring_mult x (ring_mult (ring_unit K) (field_inverse x)):K);
auto with algebra.
apply Trans with (ring_mult x (field_inverse x):K); auto with algebra.
Qed.
Hint Resolve FIELD_inverse_law: algebra.

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