Lemma in Farkas/Kra: Riemann surfaces on construction of domain satisfying certain properties

Question

I'm having some trouble understanding the proof of this lemma. I can follow the construction of $u$ and $D$, however the final step in the proof seems to be without justification. Specifically why does $D$ satisfy (3.1.2) and (3.1.3)? It's obvious from the construction that it's satisfies (3.1.1).