intros x y z H' H'0; try assumption.
apply Trans with (ring_mult y (ring_unit K)).
auto with algebra.
apply Trans with (ring_mult y (ring_mult x (field_inverse x))).
apply Sym; auto with algebra.
apply Trans with (ring_mult (ring_mult y x) (field_inverse x)).
auto with algebra.
apply Trans with (ring_mult (ring_mult z x) (field_inverse x)).
auto with algebra.
apply Trans with (ring_mult z (ring_mult x (field_inverse x))).
auto with algebra.
apply Trans with (ring_mult z (ring_unit K)).
auto with algebra.
auto with algebra.
Qed.

Lemma FIELD_inverse_non_zero :
forall x : K,
~ Equal x (monoid_unit K) -> ~ Equal (field_inverse x) (monoid_unit K).