Tableau Extension Lemma/General Statement

Theorem
Let $T$ be a finite propositional tableau.

Let its hypothesis set $\mathbf H$ be finite. Let $\mathbf H'$ be another finite set of WFFs.

Then there exists a finished finite propositional tableau $T'$ such that:

$(1):\quad$ the root of $T'$ is $\mathbf H \cup \mathbf H'$;

$(2):\quad$ $T$ is a rooted subtree of $T'$.