intuition.
apply FIELD_unit_non_zero.
apply Trans with (ring_mult (field_inverse x) x:K); auto with algebra.
apply Sym; auto with algebra.
apply Trans with (ring_mult (monoid_unit K) x:K); auto with algebra.
Qed.
Hint Resolve FIELD_inverse_non_zero: algebra.

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