Let $\mathfrak{X}^{\textup{an}}=\sum_{j}m_{j}X_{j}$ be the decomposition into prime cycles. It is enough to prove the claim for the closures $\overline{X}_{j}$ of $X_{j}$ in $\mathfrak{X}$. We may hence assume that $\mathfrak{X}^{\textup{an}}$ is irreducible and reduced. Furthermore by passing to a dominating model as in 4, we may assume that the special fibre $\tilde{\mathfrak{X}}$ of $\mathfrak{X}$ is reduced. As $\mathfrak{X}^{\textup{an}}$ has no boundary, every irreducible component of $\tilde{\mathfrak{X}}$ is proper by Corollary A.4 and hence using commutativity of the intersection product ([Gub98, Theorem 5.9]) we obtain

	$\displaystyle\int_{\mathfrak{X}^{\textup{an}}}f_{0}\;c_{1}(\mathfrak{L}_{1})\wedge...\wedge c_{1}(\mathfrak{L}_{n})$	$\displaystyle=\sum_{\begin{subarray}{c}Y\in\irr(\tilde{\mathfrak{X}})\end{subarray}}f_{0}(\zeta_{Y})\cdot\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}(Y)$
		$\displaystyle=\Deg_{\mathfrak{L}_{1},...,\mathfrak{L}_{n}}\left(\cyc(\Div_{\mathfrak{L}_{0}}(1))\right)$
		$\displaystyle=\Deg_{\mathfrak{L}_{0},\mathfrak{L}_{2},...,\mathfrak{L}_{n}}\left(\cyc(\Div_{\mathfrak{L}_{1}}(1))\right)$
		$\displaystyle=\sum_{\begin{subarray}{c}Y\in\irr(\tilde{\mathfrak{X}})\end{subarray}}f_{1}(\zeta_{Y})\cdot\Deg_{\mathfrak{L}_{0},\mathfrak{L}_{2},...,\mathfrak{L}_{n}}(Y)$
		$\displaystyle=\int_{\mathfrak{X}^{\textup{an}}}f_{1}\;c_{1}(\mathfrak{L}_{0})\wedge c_{1}(\mathfrak{L}_{2})\wedge...\wedge c_{1}(\mathfrak{L}_{n}).$

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