We write $v=\sum_{j\in J}s_{j}e_{j}$ with $s_{j}>0$ rational and $\sum_{j\in J}s_{j}=1$. Set $s_{i}=0$ for $i\in I\setminus J$. For $i\in I$ let $\varphi_{i}$ be the model function induced by the vertical divisor $b_{i}E_{i}\in\Div_{0}(\mathcal{X})$. This function is affine on each face of $\Delta$ and satisfies $\varphi_{i}(e_{j})=\delta_{ij}$ for all $j\in I$. Since $e^{\prime}_{j}=\varepsilon e_{j}+(1-\varepsilon)v$ for $j\in L$ we get:

$$\varphi_{i}(e^{\prime}_{j})=\begin{cases}\varepsilon+(1-\varepsilon)s_{i}&\text{if $i=j\in J$}\\ (1-\varepsilon)s_{i}&\text{if $i\neq j\in J$}\\ \varepsilon&\text{if $i=j\in L\setminus J$}\\ 0&\text{if $i\neq j\in L\setminus J$}\end{cases}$$

By Theorem 3.11, $E^{\prime}_{j}$ intersects $E^{\prime}_{J}$ iff $j\in L$. We thus have

$$\rho^{*}(b_{i}E_{i})|_{E^{\prime}_{J}}=\sum_{j\in L}\varphi_{i}(e^{\prime}_{j})b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\ \text{for all $i\in I$}$$

and

$$(\mu_{*}G-\varphi(v)\rho^{*}\mathcal{X}_{0})|_{E^{\prime}_{J}}=\sum_{j\in L}(\varphi(e^{\prime}_{j})-\varphi(v))b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}$$

in $\Pic(E^{\prime}_{J})_{\mathbf{Q}}$, where we have set $b^{\prime}_{j}:=\ord_{E^{\prime}_{j}}(t)$.

Recall also that $\varphi$ is affine on each segment $[v,e^{\prime}_{i}]$, so that $D_{v}\varphi(e_{i})=\varepsilon^{-1}\left(\varphi(e^{\prime}_{i})-\varphi(v)\right)$ for $i\in L$. We can now compute in $\Pic(E^{\prime}_{J})_{\mathbf{Q}}$

$$\left.\rho^{*}\left(\sum_{i\in L}D_{v}\varphi(e_{i})\,b_{i}E_{i}\right)\right|_{E^{\prime}_{J}}=\sum_{i\in L}\varepsilon^{-1}\left(\varphi(e^{\prime}_{i})-\varphi(v)\right)\left(\sum_{j\in L}\varphi_{i}(e^{\prime}_{j})\,b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\right)=\\ =\sum_{i\in J}\varepsilon^{-1}(\varphi(e^{\prime}_{i})-\varphi(v))\left(\varepsilon\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}+s_{i}\sum_{j\in J}(1-\varepsilon)\,b^{\prime}_{j}\,E^{\prime}_{j}|_{E^{\prime}_{J}}\right)+\\ +\sum_{i\in L\setminus J}\varepsilon^{-1}(\varphi(e^{\prime}_{i})-\varphi(v))\,\varepsilon\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}=\\ =\sum_{i\in L}(\varphi(e^{\prime}_{i})-\varphi(v))\,b^{\prime}_{i}\,E^{\prime}_{i}|_{E^{\prime}_{J}}+\varepsilon^{-1}(1-\varepsilon)\left(\sum_{i\in J}s_{i}(\varphi(e^{\prime}_{i})-\varphi(v))\right)\left(\sum_{j\in J}\,b^{\prime}_{j}\,{E^{\prime}_{j}}|_{E^{\prime}_{J}}\right)\\ =(\mu_{*}G-\varphi(v)\rho^{*}\mathcal{X}_{0})|_{E^{\prime}_{J}}.$$

The last equality follows from the fact that $\varphi$ is affine on the simplex $\sigma^{\prime}_{J}$ of $\Delta^{\prime}$ so that $\sum_{i\in J}s_{i}\varphi(e^{\prime}_{i})=\varphi(v)=\sum_{i\in J}s_{i}\varphi(v)$. This concludes the proof. ∎