Pick a common determination $\mathcal{X}$ of $f,g$ and the $\theta_{i}$’s, and divisors $D,D^{\prime},D_{i}$ such that $f=\varphi_{D}$, $g=\varphi_{D^{\prime}}$ and $\theta_{i}$ is the class in $N^{1}(\mathcal{X}/S)$ induced by $D_{i}$. Then by definition we have

$$\int f\,dd^{c}g\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}=\sum_{E}\ord_{E}(D)(D^{\prime}|_{E}\cdot D_{1}|_{E}\cdot...\cdot D_{n-1}|_{E})\\ =\sum_{E,E^{\prime}}\ord_{E}(D)\ord_{E^{\prime}}(D^{\prime})\,(D_{1}|_{E})|_{E^{\prime}\cap E}\cdot...\cdot(D_{n-1}|_{E})|_{E^{\prime}\cap E}\\ =\sum_{E,E^{\prime}}\ord_{E}(D)\ord_{E^{\prime}}(D^{\prime})\,(D_{1}|_{E^{\prime}})|_{E\cap E^{\prime}}\cdot...\cdot(D_{n-1}|_{E^{\prime}})|_{E\cap E^{\prime}}\\ =\int g\,dd^{c}f\wedge\theta_{1}\wedge\dots\wedge\theta_{n-1}$$

where the third equality follows from [Ful98, Theorem 2.4]. ∎