I am struggling to understand the second part of the following proof of Theorem 1.4 of the book "Condenser Capacities and Symmetrization in Geometric Function Theory" by Vladimir N. Dubinin (in what follows $U(z_0,r) \subset \mathbb{C}$ is the open disk of radius $r > 0$ and centre $z_0$, $\text{Lip}(S)$ refers to the Lipschitz continuous real-valued functions on the set $S \subset \mathbb{C}$ and all the results referred to in the proof of Theorem 1.4 have been provided below the proof of Theorem 1.4):
$\textbf{Theorem 1.4 and proof}$
That is, I do not understand what the author means by "Repeating the above reasoning for each connected component and taking account of Theorem 1.3 we obtain $v(\phi(z)) \in \text{Lip}(\overline{U(z_0,r)})$" where this sentence occurs in the image below:
The results used in the proof of Theorem 1.4 are Theorem 1.3 and Lemma 1.1, and Theorem 1.1 which are given below for reference.
$\textbf{Lemma 1.1 and proof}$
$\textbf{Theorem 1.3 and proof}$
$\textbf{Theorem 1.1 and proof}$ Theorem 1.1 says that a function $v$ defined on a compact planar set $E$ is Lipschitz on $E$ if and only if it is locally Lipschitz on $E$, that is if and only if there is a collection of open sets $U_{i}$ covering $E$, such that $v \in \text{Lip}(U_{i} \cap E)$ for every $i$.
$\textbf{What I think is going on:}$ What seems to be happening is that each connected component $C_j$ of $U(z_0,r) \cap B$ should contain $z_0$ on its boundary, and $z_0$ should be contained on an analytic arc of $\partial B$ which forms part of the boundary of $C_j$. Therefore we should be in an analogous situation to when we assumed $U(z_0,r) \cap B$ were connected, because under these assumptions we should be able to extend $\phi \restriction C_j$ analytically to a small open disk around $z_0$ for each connected component $C_j$ of $U(z_0,r) \cap B$, and then apply Theorem 1.3 as the author implies.
$\textbf{Where I have difficulty understanding:}$: However, there are several things that I am totally lost on. The first of which being, how do we know for sure that each connected component $C_j$ of $U(z_0,r) \cap B$ has as part of its boundary, an analytic arc containing $z_0$ (more precisely, it seems it is possible to lose analyticity at $z_0$, e.g. if two curves at $z_0$ form the boundary and have different tangent vectors at $z_0$)? Moreover, what is a suitable definition of "boundary consist of finitely many piecewise analytic curves" (I am assuming the definition coincides with the definition I provide in $\textbf{Edit 3}$ of this post).
It would be great if anyone could shed light on why what the author seems to be doing can be done.