# The image of annuli of the non-Archimedean projective line by rational functions

I'm reading the book "potential theory and dynamics over the Berkovich projective line" by Baker and Rumely. The proposition 2.18 in this claims that if you choose suitable finite $\{a_i\} \in D(a,r)$ for any $a\in K$(: algebraically closed complete valuation field), $r\in \mathbb{R}$ and $\phi\in K[T]$, there exist a finite number of $\{b_j\}, b \in K$ and $R \in \mathbb{R}$such that $\phi(D(a,r)\setminus \cup D(a_i,r)^-)=D(b,R)\setminus \cup D(b_j,R)^-$, where $D(a,r)$ is a closed disc centered in $a$ and the radius $r$ while $D(a,r)^-$ is the open disc.

This is actually not the exact claim of the proposition but seems the only non-trivial claim without any comments which is however necessary to complete the proof. How can I prove this? The point is, of course, the radius of all removing open balls are same.