Let $\overline{L}_{0}$ be the trivial line bundle with global section $s_{0}=1$ and metric defined by $\varphi=\log\left\|{s_{0}}\right\|^{-1}$. Let $s_{1}=s$ and, for $2\leq j\leq k$, let $s_{j}$ be an invertible meromorphic section of $L_{j}$. Since $\operatorname{div}(s_{0}|_{\mathrm{Z}})=0$,

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})=\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{1})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}$$

and

$$(\mathop{\widehat{\operatorname{div}}}(s_{0})\mathop{\widehat{\operatorname{div}}}(s_{2})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\operatorname{div}(s|_{\mathrm{Z}}))=\int_{\mathrm{X}}\varphi c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\operatorname{div}(s|\mathrm{Z})}$$

One the other hand, the symmetry of the local height pairing implies that

	$\displaystyle(\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})$	$\displaystyle=(\mathop{\widehat{\operatorname{div}}}(s_{1})\mathop{\widehat{\operatorname{div}}}(s_{0})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\mathrm{Z})$
		$\displaystyle=(\mathop{\widehat{\operatorname{div}}}(s_{0})\mathop{\widehat{\operatorname{div}}}(s_{2})\dots\mathop{\widehat{\operatorname{div}}}(s_{k})|\operatorname{div}(s|_{\mathrm{Z}}))$
		$\displaystyle\hskip 85.35826pt{}+\int_{\mathrm{X}}\log\left\|{s}\right\|^{-1}c_{1}(\overline{L}_{0})c_{1}(\overline{L}_{2})\dots c_{1}(\overline{L}_{k})\delta_{\mathrm{Z}}.$

Combining these equations, we obtain the claim. ∎