We use the notation of [BPS14, §3.5]. Let $\mathscr{X}_{\Lambda}$ be the affine toric scheme associated to $\Lambda$. The ring of functions of $\mathscr{X}_{\Lambda}$ is

$$K^{\circ}[\mathscr{X}_{\Lambda}]=K^{\circ}[\widetilde{M}_{\Lambda}]/(\chi^{(0,1)}-\varpi).$$

The orbit $O(\Lambda)$ is a closed subscheme of $\mathscr{X}_{\Lambda}$. If $u\in\rint(\Lambda)$, the ideal of $O(\Lambda)$ is the ideal generated by the monomials $\chi^{(m,l)}$ with $(m,l)\in\widetilde{M}_{\Lambda}$ and $\langle m,u\rangle+l>0$.

The generic fiber of $\mathscr{X}_{\Lambda}$ is the affine toric variety $X_{\rec(\Lambda)}=\Spec(K[M_{\rec(\Lambda)}])$. The natural inclusion $K^{\circ}[\mathscr{X}_{\Lambda}]\subset K[M_{\rec(\Lambda)}]$ is given by $\chi^{(m,l)}\mapsto\varpi^{l}\chi^{m}$. Any point $p\in X_{\rec(\Lambda)}^{{\mathrm{an}}}$ determines a seminorm on $K^{\circ}[\mathscr{X}_{\Lambda}]$. The set of points of $X^{{\mathrm{an}}}_{\rec(\Lambda)}$ whose reduction belongs to $\mathscr{X}_{\Lambda}$ is

$$C=\{p\in X^{{\mathrm{an}}}_{\rec(\Lambda)}\mid|f(p)|\leq 1,\ \forall f\in K^{\circ}[\mathscr{X}_{\Lambda}]\}.$$

Given a point $p\in C$, then ${\mathrm{red}}(p)$ is the point corresponding to the prime ideal

$$\mathfrak{q}_{p}=\{f\in K^{\circ}[\mathscr{X}_{\Lambda}]\mid|f(p)|<1\}.$$

Every $f\in K^{\circ}[\mathscr{X}_{\Lambda}]$ can be written as a sum

$$f=\sum_{(m,l)\in\widetilde{M}_{\Lambda}}\alpha_{(m,l)}\chi^{(m,l)},$$

with $|\alpha_{(m,l)}|=0,1$ and only a finite number of coefficients $\alpha_{(m,l)}$ different from zero.

By the definition of $\zeta_{K}$,

	$\displaystyle\mathfrak{q}_{\zeta_{K}(u)}$	$\displaystyle=\left\{\sum\alpha_{(m,l)}\chi^{(m,l)}\in K^{\circ}[\mathscr{X}_{\Lambda}]\,\middle|\,|\varpi|^{l}e^{-\lambda_{K}\langle m,u\rangle}<1\right\}$
		$\displaystyle=\left\{\sum\alpha_{(m,l)}\chi^{(m,l)}\in K^{\circ}[\mathscr{X}_{\Lambda}]\,\middle|\,\langle m,u\rangle+l>0\right\}.$

Since $u\in\rint(\Lambda)$, we deduce that $\mathfrak{q}_{\zeta_{K}(u)}$ is the ideal of $O(\Lambda)$ and therefore ${\mathrm{red}}(\zeta_{K}(u))=\xi_{\Lambda}.$ ∎