This week I will focus on the following theorem. Let M be Mitchell's partial ordering from the previous two talks. Working in V^M, let Q be either ccc forcing of size aleph1 or Add(omega,mu) for some infinite mu. Then the tree property holds at omega2 in V^{M*Q}. The theorem is essentially an application of a new branch lemma. In this talk, I will sketch a proof of both the theorem and the lemma.