Finished Propositional Tableau has Finished Branch or is Confutation

Theorem
Let $\left({T, \mathbf H, \Phi}\right)$ be a finished propositional tableau.

Then one of the following holds:


 * $T$ has a finished branch;
 * $T$ is a confutation.

Proof
Suppose $T$ has no finished branches.

Since $T$ is finished, it follows that every branch of $T$ is contradictory.

That is, $T$ is a confutation.