Lemma 1.5.3. [04MU]

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Lemma 1.5.3.

Let $x\in\mathbb{T}^{\an}$ . Then $x$ has a center on $\mathscr{X}$ if and only if $\zeta(\val(x))$ has a center on $\mathscr{X}$ . Moreover, if this holds, we have $c_{{\mathscr{X}}}(x)\in\overline{c_{{\mathscr{X}}}(\zeta(\val(x)))}$ .