(a) State Zorn's Lemma, and use it to prove that every nontrivial distributive lattice $L$ admits a lattice homomorphism $L \rightarrow\{0,1\}$.

(b) Let $S$ be a propositional theory in a given language $\mathcal{L}$. Sketch the way in which the equivalence classes of formulae of $\mathcal{L}$, modulo $S$-provable equivalence, may be made into a Boolean algebra. [Detailed proofs are not required, but you should define the equivalence relation explicitly.]

(c) Hence show how the Completeness Theorem for propositional logic may be deduced from the result of part (a).