Explain what it means for a poset to be chain-complete. State Zorn's Lemma, and use it to prove that, for any two elements $a$ and $b$ of a distributive lattice $L$ with $b \not{\leftarrow} a$, there exists a lattice homomorphism $f: L \rightarrow\{0,1\}$ with $f(a)=0$ and $f(b)=1$. Explain briefly how this result implies the completeness theorem for propositional logic.