Lemma 33.45.6. In the situation of Definition 33.45.3 let $Z_ i \subset Z$ be the irreducible components of dimension $d$. Let $m_ i = \text{length}_{\mathcal{O}_{X, \xi _ i}}(\mathcal{O}_{Z, \xi _ i})$ where $\xi _ i \in Z_ i$ is the generic point. Then

\[ (\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z) = \sum m_ i(\mathcal{L}_1 \cdots \mathcal{L}_ d \cdot Z_ i) \]

**Proof.**
Immediate from Lemma 33.45.2 and the definitions.
$\square$

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