Proof verification for $\vdash_{TA} (p \rightarrow q) \rightarrow ((r \rightarrow p) \rightarrow (r \rightarrow p))$

Question

I need to prove that using analytics tableaux: $\vdash (p \rightarrow q) \rightarrow ((r \rightarrow p) \rightarrow (r \rightarrow p))$.

I did the following:

I'm confused because $Tp \rightarrow q$ is a branchable expression, but if I did branch it, it would be open. But if that is right, I could have closed in line 5, cause 3 and 4 are already contradictions and that would also be enough, right?

I asked a similar question before, which involved an atom, but in this case, it is a potentially branchable expression that is not being used.