Tableau Confutation is Finished

Theorem
Let $T$ be a propositional tableau.

Let $T$ be a tableau confutation.

Then $T$ is a finished tableau.

Proof
By definition of tableau confutation, $T$ is a finite propositional tableau.

From Branch of Finite Propositional Tableau is Finite, every branch of $T$ is finite.

Also by definition of tableau confutation, every branch of $T$ is contradictory.

The result follows by definition of finished propositional tableau.