Let $p\in\operatorname{im}\theta_{\Sigma}$ be such that ${\operatorname{val}}_{K}(p)>a$, hence $|t(p)|<|\varpi|^{a}$. By Lemma 5.57, this implies that ${\operatorname{red}}(p)=q_{0}$. In a neighbourhood of $q_{0}$, the divisor of the rational section $s_{\Psi}^{\otimes e}t^{em_{0}}\varpi^{-\alpha_{0}-em_{0}a}$ is zero, and so

$$\|s_{\Psi}^{\otimes e}(p)t^{em_{0}}(p)\varpi^{-\alpha_{0}-em_{0}a}\|=1.$$

Set $u={\operatorname{val}}(p)$. Then,

	$\displaystyle\psi_{\|\cdot\|}(u)$	$\displaystyle=\frac{\log\|s_{\Psi}^{\otimes e}(p)\|}{e\lambda_{K}}$
		$\displaystyle=\frac{-em_{0}\log|t(p)|+(\alpha_{0}+em_{0}a)\log|\varpi|}{-e\log|\varpi|}$
		$\displaystyle=m_{0}(u-a)-\frac{\alpha_{0}}{e}.$

The other cases are proved in a similar way. ∎