Since the metric is algebraic, there exist a proper $K^{\circ}$- scheme $\mathcal{X}$ and a line bundle $\mathcal{L}$ on $\mathcal{X}$ such that the base change of $(\mathcal{X},\mathcal{L})$ to $K$ is isomorphic to $(X_{\Sigma},L^{\otimes e})$. Let $\{{\mathcal{U}}_{i},s_{i}\}$ be a trivialization of $\mathcal{L}$. Let $C_{i}={\operatorname{red}}^{-1}({\mathcal{U}}_{i}\cap\mathcal{X}_{o})$. The subsets $C_{i}$ form a finite closed cover of $X_{\Sigma}^{{\text{\rm an}}}$. On ${\mathcal{U}}_{i}$ we can write $s_{\Psi}^{\otimes e}=g_{i}s_{i}$ for certain rational function $g_{i}$. Therefore, on $C_{i}$, we have $\log\|s_{\Psi}(p)\|=\frac{\log|g(p)|}{e}$. By Lemma 5.48, it follows that there is a finite closed cover of $N_{\mathbb{R}}$ and the restriction of $\psi_{\|\cdot\|}$ to each of these closed subsets is rational piecewise affine. Therefore $\psi_{\|\cdot\|}$ is rational piecewise affine. The second statement follows from the first and Corollary 5.47. ∎