intros...
assert (a * b / b = a)...
field...
intuition...
assert (a * b / b = 0 / b)...
rewrite H1 in H3...
assert (a = 0)...
assert (0 / b = 0)...
field...
Qed.

Lemma nonzerodiv : forall a b : F, a <> 0 -> b <> 0 -> a / b <> 0.
Proof with try solve [ auto with field_hints | congruence ].