I have 2 versions of the compactness theorem;
Compactness Theorem $1$ - Suppose that $S$ is a set of propositional terms. Then $S$ is satisfiable if and only if every finite subset $S' \subseteq$ $S$ is satisfiable.
Compactness Theorem $2$ - Suppose that $S$ is a set of terms and that $t$ is a term. Then $S \vDash t$ iff there is a finite set $ S' \subseteq S$ such that $S' \vDash t$.
Now, I'm trying to figure out how you can get from one version to the other, but I'm not sure how to formally argue it. I can understand it intuitively, but I don't know how to formulate the argument at all. How can I relate the set $S$ being satisfiable to logical consequences $ S \vDash t$ etc.
Any help would be appreciated!