For short, we set $\Delta=\Delta_{\Psi}$ and $\psi=\psi_{\|\cdot\|}$. Let $\Sigma_{\Delta}$ be the fan associated to $\Delta$ as in Remark 4.43. There is a toric morphism $\varphi\colon X_{\Sigma}\to X_{\Sigma_{\Delta}}$. The function $\psi^{\vee}$ defines an approachable metric $\|\cdot\|^{\prime}$ on $\mathcal{O}(D_{\Psi_{\Delta}})^{{\text{\rm an}}}$. We denote ${\overline{L}}^{\prime}=(\mathcal{O}(D_{\Psi_{\Delta}}),\|\cdot\|^{\prime})$. Then there is an isometry $\varphi^{\ast}({\overline{L}}^{\prime})={\overline{L}}$. By Corollary 5.25 there is an isometry $\varphi^{\ast}({\overline{L}}^{\prime}{}^{{\operatorname{can}}})={\overline{L}}^{{\operatorname{can}}}$.

If the dimension of $\Delta$ is less than $n$, then the right-hand side of equation (6.7) is zero. Moreover, $n=\dim(X_{\Sigma})>\dim(X_{\Sigma_{\Delta}})$ and the metrized line bundles ${\overline{L}}$ and ${\overline{L}}^{{\operatorname{can}}}$ come from a variety of smaller dimension. Therefore, by Theorem 2.46(2), the left-hand side of equation (6.7) is also zero, because $\varphi_{*}X_{\Sigma}$ is the cycle zero. If $\Delta$ has dimension $n$ then $\varphi$ is a birational morphism, so, by Theorem 2.46(2),

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}^{\prime}}(X_{\Sigma_{\Delta}}).$$

Therefore it is enough to prove the theorem for $X_{\Sigma_{\Delta}}$. By construction, the fan $\Sigma_{\Delta}$ is regular; hence the variety $X_{\Sigma_{\Delta}}$ is projective and $L^{\prime}$ is ample. Thus we are reduced to prove the theorem in the case when $\Sigma$ is regular and $L$ is ample.

Now the proof is done by induction on $n$, the dimension of $X_{\Sigma}$. If $n=0$ then $X_{\Sigma}=\mathbb{P}^{0}$, $\Psi=0$, $\Delta=\{0\}$ and $L=\mathcal{O}(D_{0})=\mathcal{O}_{\mathbb{P}^{0}}$. By equation (5.15), $\log\|s\|=\lambda_{K}\psi(0)$ and $\log\|s\|_{{\operatorname{can}}}=\lambda_{K}\Psi(0)=0$. The Legendre-Fenchel dual of $\psi$ satisfies $\psi^{\vee}(0)=-\psi(0)$. By equation (2.40), $\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s)=-\lambda_{K}\psi(0)$ and $\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(X_{\Sigma};s)=0$. Therefore

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=-\lambda_{K}\psi(0)=\lambda_{K}\psi^{\vee}(0)=1!\lambda_{K}\int_{\Delta}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M}.$$

Let $n\geq 1$ and let $s_{0},\dots,s_{n-1}$ be rational sections of ${\mathcal{O}}(D_{\Psi})$ such that $s_{0},\dots,s_{n-1},s$ intersect $X_{\Sigma}$ properly. By the construction of local heights (Definition 2.39),

(6.10)		$\displaystyle\operatorname{h}_{{\overline{L}}}(X_{\Sigma};s_{0},\dots,s_{n-1},s)=\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);$	$\displaystyle s_{0},\dots,s_{n-1})$
		$\displaystyle-\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}$

and a similar formula holds for the canonical metric.

For each facet $F$ of $\Delta$ let $v_{F}$ be as in Notation 3.103. Since $L$ is ample, Proposition 4.46 implies

(6.11)

$$\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);s_{0},\dots,s_{n-1})=\sum_{F}-\langle v_{F},F\rangle\operatorname{h}_{{\overline{L}}}(V(\tau_{F});s_{0},\dots,s_{n-1}),$$

where the sum is over the facets $F$ of $\Delta$. Observe that the local height of $V(\tau_{F})$ with respect to the metrized line bundle ${\overline{L}}$ coincides with the local height associated to the restriction of ${\overline{L}}$ to this subvariety. Moreover by Corollary 5.23, the restriction of the canonical metric of $L^{{\text{\rm an}}}$ to this subvariety agrees with the canonical metric of $L^{{\text{\rm an}}}|_{V(\tau_{F})}$. Hence, by substracting from equation (6.11) the analogous formula for the canonical metric, we obtain

(6.12)		$\displaystyle\sum_{F}-\langle m_{F},v_{F}\rangle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}|_{V(\tau_{F})}}(V(\tau_{F}))=\operatorname{h}_{{\overline{L}}}(\operatorname{div}(s);$	$\displaystyle s_{0},\dots,s_{n-1})$
		$\displaystyle-\operatorname{h}_{{\overline{L}}^{{\operatorname{can}}}}(\operatorname{div}(s);s_{0},\dots,s_{n-1}).$

Moreover, Proposition 2.37 implies that

$$\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=\int_{X_{\Sigma,0}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}.$$

By equation (5.15), $\log\|s\|=({\operatorname{val}}_{K})^{*}(\lambda_{K}\psi)$. Moreover

$\displaystyle\int_{X_{\Sigma,0}^{{\text{\rm an}}}}({\operatorname{val}}_{K})^{*}(\lambda_{K}\psi)c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=$

$\displaystyle\int_{N_{\mathbb{R}}}\lambda_{K}\psi({\operatorname{val}}_{K})_{*}(c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}})$

and by Theorem 5.81, $({\operatorname{val}}_{K})_{*}(c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}})=n!{\mathcal{M}}_{M}(\psi)$. Hence

(6.13)

$$\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|c_{1}({\overline{L}})^{n}\wedge\delta_{X_{\Sigma}}=n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi).$$

By Example 3.96, ${\mathcal{M}}_{M}(\Psi)=\operatorname{vol}_{M}(\Delta)\delta_{0}$. Therefore, in the case of the canonical metric, equation (6.13) reads as

(6.14)

$$\int_{X_{\Sigma}^{{\text{\rm an}}}}\log\|s\|_{{\operatorname{can}}}c_{1}({\overline{L}}^{{\operatorname{can}}})^{n}\wedge\delta_{X_{\Sigma}}=n!\lambda_{K}\operatorname{vol}_{M}(\Delta)\Psi(0)=0.$$

Thus, substracting from equation (6.10) the analogous formula for the canonical metric and using equations (6.12), (6.13) and (6.14), we obtain

(6.15)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})=\sum_{F}-\langle v_{F},F\rangle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}|_{V(\tau_{F})}}(V(\tau_{F}))-n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi).$$

By the inductive hypothesis and equation (5.77)

$$\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}|_{V(\tau_{F})}}(V(\tau_{F}))=n!\lambda_{K}\int_{F}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M({F})}.$$

Hence, by Corollary 3.104,

	$\displaystyle\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(X_{\Sigma})$	$\displaystyle=-n!\lambda_{K}\sum_{F}\langle v_{F},F\rangle\int_{F}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M({F})}-n!\lambda_{K}\int_{N_{\mathbb{R}}}\psi{\mathcal{M}}_{M}(\psi)$
		$\displaystyle=(n+1)!\lambda_{K}\int_{\Delta}\psi^{\vee}\,\text{\rm d}\operatorname{vol}_{M},$

proving the theorem ∎