Branch of Finite Propositional Tableau is Finite
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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 $T$ is formed by applying the tableau extension rules a finite number of times.
Each tableau extension rule extends $T$ finitely.
Therefore $T$ is a finite tree.
The result follows from Branch of Finite Tree is Finite.
$\blacksquare$