# Prove that $K$ is finitely definable iff $K$ has finite support

Hi guys I need to prove a Finite Support Theorem which states that $K$ is finitely definable iff $K$ has finite support. Unfortunately I succeeded in proving only the first part of if and only if.

$Introduction:$ We say that $S\subseteq \{p_i : i\in\mathbb{N}\}$ is a support for a set of assignments $K$, if for every two assignments $v$ and $v'$ that agree on all propositional variables in $S$ $\Rightarrow$ $v$$\in$$K$$\iff$$v'$$\in$$K$

$K$ is finitely definable if there exists a finite number of formulas in $\Sigma$ s.t. $Ass(\Sigma) = K$

$Proof \ attempt:$

Let $K$ be a set of truth assignments.

$\Rightarrow$ Suppose $K$ is finitely definable, I can just go over all the formulas in $\Sigma$ because it is finite and construct a finite set $S$ from a propositional variables in $\Sigma$, with some formality it's not a problem to prove that $S$ is a finite support for $K$.

$\Leftarrow$ Suppose $S$ is a finite support of $K$. I need to show that $K$ is finitely definable. In this direction I'm struggling to find a formal proof. Maybe I need somehow to construct $\Sigma$ from the a finite support I have, but I don't know how to use the given property of assignments... Anyway I'm a little lost here. Would be thankful for some help.