Let $\|\cdot\|$ be a toric metric on $L^{{\text{\rm an}}}$. Since $s$ is a regular nowhere vanishing section on $X_{0}^{{\text{\rm an}}}$, $\psi_{\|\cdot\|}$ is a well defined continuous function on $N_{\mathbb{R}}$. Let $\{m_{\sigma}\}$ be a set of defining vectors of $\Psi$. For each cone $\sigma\in\Sigma$, the section $\chi^{m_{\sigma}}s$ is a regular nowhere vanishing section on $X_{\sigma}^{{\text{\rm an}}}$. Therefore $\log(\|(\chi^{m_{\sigma}}s)(p)\|)$ is a continuous function on $X_{\sigma}^{{\text{\rm an}}}$ that is $\mathbb{S}^{{\text{\rm an}}}$-invariant. So it defines a continuous function on $X_{\sigma}(\mathbb{R}_{\geq 0})$. By equation (5.4),

Therefore $\psi_{\|\cdot\|}-m_{\sigma}$ extends to a continuous function on $N_{\sigma}\simeq X_{\sigma}(\mathbb{R}_{\geq 0})$. If we see that $\Psi-m_{\sigma}$ extends also to a continuous function on $N_{\sigma}$ we will be able to extend $\psi_{\|\cdot\|}-\Psi$ to a continuous function on $N_{\sigma}$ for every $\sigma\in\Sigma$ and therefore to $N_{\Sigma}$.

Let $\tau$ be a face of $\sigma$ and let $u\in N(\tau)_{\mathbb{R}}$. Let $W(\tau,U,p)$ be a neighbourhood of $u$ as in (5.6). By taking $U$ small enough and $p$ big enough we can assume that $W(\tau,U,p)\cap N_{\mathbb{R}}$ is contained in the set of cones that have $\tau$ as a face. Since $\Psi$ and $m_{\sigma}$ agree when restricted to $\sigma$ (hence when restricted to $\tau$) it follows that, if $w+t\in W(\tau,U,p)\cap N_{\mathbb{R}}$ with $w\in U$ and $t\in p+\tau$, then $(\Psi-m_{\sigma})(w+t)$ only depends on $w$ and not on $t$. Hence it can be extended to a continuous function on the whole $W(\tau,U,p)$. By moving $\tau$, $u$, $U$ and $p$ we see that it can be extended to a continuous function on $N_{\sigma}$.

Let now $\psi$ be a function on $N_{\mathbb{R}}$ such that $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$. We define a toric metric $\|\cdot\|_{\psi}$ on $L^{{\text{\rm an}}}$ over the set $X_{0}^{{\text{\rm an}}}$ by the formula

Then, by the argument before, $\psi-m_{\sigma}$ extends to a continuous function on $N_{\sigma}$, which proves that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\sigma}^{{\text{\rm an}}}$. Varying $\sigma\in\Sigma$ we obtain that $\|\cdot\|_{\psi}$ extends to a metric over $X_{\Sigma}^{{\text{\rm an}}}$. ∎