Let $\mathcal{X}$ be an SNC model such that all $x_{i}$ appear as vertices of $\Delta_{\mathcal{X}}$. By Theorem 2.10 there exists a constant $M>0$ such that $\sup_{X}\varphi\leq M$ for all $\varphi\in\PSH(X,\omega)$ such that $\varphi(x_{1})\leq t_{1}$. Since adding a constant $c$ to the $t_{i}$ only replaces $\varphi_{S,t}$ with $\varphi_{S,t}+c$, we may thus assume $t_{i}\leq-1$ and $\varphi\leq-1$ as soon as $\varphi\in\PSH(X,\omega)$ satisfies $\varphi(x_{1})\leq t_{1}$. Now let $f_{\mathcal{X}}\in\mathcal{D}(X)_{\mathbf{R}}$ be the unique function that is linear on the faces of $\Delta_{\mathcal{X}}$, takes value $t_{i}$ at $x_{i}$ for each $i$, $0$ at any other vertex of $\Delta_{\mathcal{X}}$, and such that $f_{\mathcal{X}}=f_{\mathcal{X}}\circ p_{\mathcal{X}}$. Since each $\varphi\in\PSH(X,\omega)$ is convex on the faces of $\Delta_{\mathcal{X}}$ and satisfies $\varphi\leq\varphi\circ p_{\mathcal{X}}$, we have $\varphi(x_{i})\leq t_{i}$ for all $i$ iff $\varphi\leq f_{\mathcal{X}}$, hence $\varphi_{S,t}=P_{\omega}(f_{\mathcal{X}})$. This already shows that $\varphi_{S,t}$ is continuous and $\omega$-psh, and the orthogonality property further shows that $\MA(\varphi_{S,t})$ is supported in $\{f_{S,t}=f_{\mathcal{X}}\}$ for each SNC model $\mathcal{X}$ as above. We thus see that $\supp\MA(\varphi_{S,t})\subset\bigcap_{\mathcal{X}}\{f_{\mathcal{X}}<0\}$.

We claim that the latter intersection is in fact equal to $\{x_{1},...,x_{N}\}$, which will conclude the proof of the lemma. For each model $\mathcal{X}$ and each $x\in X$ we may consider the center (or reduction) $c_{\mathcal{X}}(x)\in\mathcal{X}_{0}$. Let $E_{i}\in\Div_{0}(\mathcal{X})$ be the component of $\mathcal{X}_{0}$ with generic point $c_{\mathcal{X}}(x_{i})$, and let $\varphi_{\mathcal{X},i}$ be the model function determined by $E_{i}$. For each $x\in X$ we have $f_{\mathcal{X}}(x)=f_{\mathcal{X}}(p_{\mathcal{X}}(x))=\sum_{i}t_{i}\varphi_{\mathcal{X},i}(x)$, hence

∎