The proof uses the toroidal theory of [KKMS] together with elementary ramification theory of valuations [ZS75].

Let $\sigma$ be the face of $\Delta({\mathcal{X}})$ corresponding to an irreducible component $Y$ of $E_{0}\cap\dots\cap E_{p}$. Set $b_{i}=\operatorname{ord}_{E_{i}}({t})$. With the identification

the integral affine structure $M_{\sigma}$ is given by the lattice ${\mathbb{Z}}^{p+1}$. Note that $b_{\sigma}=\gcd_{i}b_{i}$.

Given a closed point $\xi\in\mathring{Y}$, we can find local coordinates $z_{0},\dots,z_{n}$ in the formal completion $\widehat{\mathcal{O}}_{{\mathcal{X}},\xi}\simeq k[\![z_{0},\dots,z_{n}]\!]$ such that ${t}=\prod_{i=0}^{p}z_{i}^{b_{i}}$. A toric computation (cf. [KKMS, pp.98–102]) shows that $\xi$ has $\gcd(m,b_{\sigma})$ preimages $\xi^{\prime}_{\alpha}$ in ${\mathcal{X}}^{\prime}_{0}$, with ${\mathcal{X}}^{\prime}$ formally isomorphic, at each $\xi^{\prime}_{\alpha}$, to the product of ${\mathbb{A}}_{k}^{n-p}$ with the affine toric $k$-variety corresponding to the cone ${\mathbb{R}}_{+}^{p+1}\subset{\mathbb{R}}^{p+1}$ with lattice

It follows that $p^{-1}(\sigma)$ is the union of the corresponding faces $\sigma^{\prime}_{\alpha}$ of $\Delta({\mathcal{X}}^{\prime})$, each isomorphic to

with integral affine structure induced by $M^{\prime}$. Now $p$ restricts to a homeomorphism $\sigma^{\prime}_{\alpha}\overset{\sim}{\to}\sigma$ given by $w=w^{\prime}/m$. Thus $M_{\sigma^{\prime}_{\alpha}}=mp^{*}M_{\sigma}+{\mathbb{Z}}1_{\sigma^{\prime}_{\alpha}}$. This implies (i), and (ii)–(iii) easily follow.

Now note that

It remains to analyze the degree $f_{\sigma}$ of the restriction $Y_{\sigma^{\prime}_{\alpha}}\to Y_{\sigma}$. For this we use ramification theory.

The function field $F(X^{\prime})=F(X)({t}^{1/m})$ is a Galois extension of $F(X)$ of degree $m$, with Galois group $G$. For any valuation $v^{\prime}\in X^{\prime\operatorname{val}}$, we have $v^{\prime}|_{F(X)}=mp(v^{\prime})$.

Let $v\in X^{\mathrm{an}}$ be a valuation corresponding to a point $w\in\sigma$. Assume $w$ is “general” in the sense that $\dim_{\mathbb{Q}}\sum_{i=0}^{p}{\mathbb{Q}}w_{i}=p$. The point $w$ has $g_{\sigma}$ preimages $w^{\prime}_{\alpha}$ under $p$, one in each $\sigma^{\prime}_{\alpha}$, and the valuations $v^{\prime}_{\alpha}:=m^{-1}w^{\prime}_{\alpha}$ are all the extensions of $v$ to $F(X^{\prime})$. Let us compute the residue degree and ramification index of these extensions.

The residue fields of $v$ and $v^{\prime}_{\alpha}$ are exactly the function fields of $Y$ and $Y^{\prime}_{\alpha}$, respectively, so the residue degree of the extension $v^{\prime}_{\alpha}$ of $v$ is equal to $f_{\sigma}$.

The value group $\Gamma_{v}=v(F(X))$ of $v$ is given by $\Gamma_{v}=\sum_{i=0}^{p}{\mathbb{Z}}w_{i}$. Similarly, the value group of $v^{\prime}_{\alpha}$ is given by $\Gamma_{v^{\prime}_{\alpha}}=\frac{1}{m}{\mathbb{Z}}+\frac{1}{m}\sum_{i=0}^{p}{\mathbb{Z}}w^{\prime}_{i}=\frac{1}{m}{\mathbb{Z}}+\sum_{i=0}^{p}{\mathbb{Z}}w_{i}$. It follows that the ramification index of the extension $v^{\prime}_{\alpha}$ of $v$ is given by

By [ZS75, p.77] we now have $e_{\sigma}f_{\sigma}g_{\sigma}=m$, which completes the proof. ∎