Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension/Proof 2

Theorem
Let $T$ be a finitely satisfiable $\mathcal L$-theory.

There is a finitely satisfiable $\mathcal L$-theory $T'$ which contains $T$ as a subset such that for all $\mathcal L$-sentences $\phi$, either $\phi \in T'$ or $\neg\phi \in T'$.

Proof
Let $\mathcal A$ be the set of finitely satisfiable extensions of $T$.

By the lemma, for each element $S$ of $\mathcal A$ and each $\mathcal L$-sentence $\phi$, either $S \cup \{ \phi \} \in \mathcal A$ or $S \cup \{ \neg \phi \} \in \mathcal A$.

$\mathcal A$ has finite character:

If $S \in \mathcal A$ and $F$ is a finite subset of $S$, then $S$ is satisfiable and hence finitely satisfiable, and thus in $\mathcal A$.

Suppose $S$ is a theory on $\mathcal L$ and every finite subset of $S$ is finitely satisfiable.

Then in fact every finite subset of $S$ is satisfiable, so $S$ is finitely satisfiable.

Thus we see that $\mathcal A$ has finite character.

By the Restricted Tukey-Teichmüller Theorem, $\mathcal A$ has an element $T'$ such that for each $\mathcal L$-sentence $\phi$ either $\phi \in T'$ or $\neg \phi \in T'$.