State Zorn's lemma, and show how it may be deduced from the Axiom of Choice using the Bourbaki-Witt theorem (which should be clearly stated but not proved).

Show that, if $a$ and $b$ are distinct elements of a distributive lattice $L$, there is a lattice homomorphism $f: L \rightarrow\{0,1\}$ with $f(a) \neq f(b)$. Indicate briefly how this result may be used to prove the completeness theorem for propositional logic.