The tensor product $s^{\otimes e}$ defines a rational section of ${\mathcal{L}}$. Let $\Lambda\in\Pi$ and choose $m_{\Lambda}\in M$, $l_{\Lambda}\in\mathbb{Z}$ such that $e\psi|_{\Lambda}=m_{\Lambda}+l_{\Lambda}|_{\Lambda}$. Let $u\in\Lambda$ and $p\in X^{{\text{\rm an}}}_{\Sigma}$ with $u={\operatorname{val}}(p)$. Then ${\operatorname{red}}(p)\in{\mathcal{X}}_{\Lambda}$. But in ${\mathcal{X}}_{\Lambda}$ the section $\chi^{m_{\Lambda}}\varpi^{l_{\Lambda}}s^{\otimes e}$ is regular and non-vanishing. Therefore, by Definition 2.17,

$$\|\chi^{m_{\Lambda}}(p)\varpi^{l_{\Lambda}}s^{\otimes e}(p)\|_{{\mathcal{L}}}=1.$$

Thus

	$\displaystyle\psi_{\|\cdot\|_{{\mathcal{L}}}}(u)$	$\displaystyle=\frac{1}{\lambda_{K}}\log(\|s(p)\|_{{\mathcal{L}}})$
		$\displaystyle=\frac{1}{e\lambda_{K}}\log(|\chi^{-m_{\Lambda}}(p)\varpi^{-l_{\Lambda}}|)$
		$\displaystyle=\frac{1}{e}(\langle m_{\Lambda},u\rangle+l_{\Lambda})$
		$\displaystyle=\psi(u).$

Therefore $\psi$ agrees with the function associated to the metric $\|\cdot\|_{{\mathcal{L}}}$. Hence $\psi-\Psi$ extends to a continuous function on $N_{\Sigma}$ and the metric $\|\cdot\|_{\psi}$ agrees with $\|\cdot\|_{{\mathcal{L}}}$. ∎