Branch of Finite Propositional Tableau is Finite

Theorem
Let $T$ be a finite propositional tableau.

Let $\Gamma$ be a branch of $T$.

Then $\Gamma$ is a finite branch.

Proof
By definition, a finite propositional tableau is the last term $T_n$ of some finite propositional tableau chain $T_0, T_1, \ldots, T_n$.

By definition of propositional tableau chain, each $T_k$ in $\left\{{T_1, T_1, \ldots, T_n}\right\}$ is obtained from $T_k$ by applying one of the tableau extension rules at a leaf node $t$ of $T_k$.

Each tableau extension rule extends any such $T_k$ finitely.

Therefore each $T_k \in \left\{{T_1, T_1, \ldots, T_n}\right\}$ is a finite tree.

The result follows from Branch of Finite Tree is Finite.