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.


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