We first assume that $f$ is linear on each edge of $R(\mathscr{X})$ and that $D$ is a sum of rational points of $X$. Then, $\overline{L}$ corresponds to the line bundle $\mathfrak{L}$ on the model $\mathfrak{X}$ given by Equation 2.2.1. By definition, the measure $c_{1}(\mathfrak{L})$ is computed as follows. It is a sum, for all components $V_{i}$ of the special fiber, of $\deg(\mathfrak{L}|V_{i})\log\left|{\pi}\right|^{-1}$ times the Dirac measure at the corresponding point $v_{i}$ of $R(\mathfrak{X})$. In particular, it is supported by $R(\mathfrak{X})$. Then,

$$\deg(\mathfrak{L}|V_{i})=\sum_{j}n_{j}\left\{\begin{array}[]{cc}1&\text{if $D_{P_{j}}$ passes through $V_{i}$ ;}\\ 0&\text{otherwise}\end{array}\right\}+\sum_{j}f(V_{j})(V_{i},V_{j}),$$

where $(V_{i},V_{j})$ is the intersection number of the divisors $V_{i}$ and $V_{j}$. That $D_{P_{j}}$ passes through $V_{i}$ means exactly that $\rho(P_{j})=v_{i}$. Moreover, if $j\neq i$, then $(V_{i},V_{j})=m_{i,j}$ is just the number of intersection points of $V_{i}$ and $V_{j}$, while

$$(V_{i},V_{i})=(V_{i},\sum_{j}V_{j})-\sum_{j\neq i}(V_{i},V_{j})=-\sum_{j\neq i}(V_{i},V_{j}),$$

since the whole special fiber is numerically equivalent to zero. Consequently,

$$\sum_{j}f(v_{j})(V_{i},V_{j})=\sum_{j\neq i}m_{i,j}\big(f(V_{j})-f(V_{i})\big).$$

Observe that this is the sum, over all edges from $V_{i}$, of the derivative of $f$ along this edge. Comparing with the definitions given by Zhang in [57], one finds, for any function $g$ on $R(\mathfrak{X})$

	$\displaystyle\sum_{i}\deg(\mathfrak{L}|V_{i})g(v_{i})$	$\displaystyle=\sum_{j}n_{j}g(\rho(P_{j}))+\sum_{i}\langle\delta f(v_{i}),g\rangle$
		$\displaystyle=\int_{R(\mathfrak{X})}g\,(\mu_{D}+\delta f)$
		$\displaystyle=\int_{R(\mathfrak{X})}g\,{\operatorname{curv}}(\overline{\mathscr{O}(D+f)}).$

This proves the claimed formula when $f$ is linear on each edge of $\mathfrak{X}$ and $D$ is a sum of rational points.

By working over an appropriate finite extension of $K$, it extends to the case where $f$ is only piecewise linear, $D$ being any divisor on $X$.

Zhang defines $\overline{\mathscr{O}(D+f)}_{\mathfrak{X}}$ to be semi-positive if $f$ is uniform limit of piecewise linear functions $f_{n}$ such that ${\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})}_{\mathfrak{X}})\geq 0$. The metrized line bundle $\overline{L}$ is then the limit of the metrized line bundles $\overline{L}_{n}$ corresponding models $\mathfrak{L}_{n}$ (on appropriate models $\mathfrak{X}_{n}$ of $\mathrm{X}$ after some extension of scalars) of $\mathscr{O}(D)$. By the previous computation, these metrics are smooth and $c_{1}(\overline{L}_{n})\geq 0$. Reversing the computation, this means that $\mathfrak{L}_{n}$ is numerically effective on $\mathfrak{X}_{n}$, hence $\overline{L}$ is semi-positive. By definition of the measure $c_{1}(\overline{L})$, one has

	$\displaystyle c_{1}(\overline{L})$	$\displaystyle=\lim_{n}c_{1}(\overline{L}_{n})=\lim_{n}\iota_{*}{\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})})$
		$\displaystyle=\iota_{*}\lim_{n}{\operatorname{curv}}(\overline{\mathscr{O}(D+f_{n})})=\iota_{*}{\operatorname{curv}}(\overline{\mathscr{O}(D+f)}).$

The case of an admissible metrized line bundle follows by linearity. ∎