Proper pushforward

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I am studying a some intersection theory. Now I am stuck on the proof of the Lemma 02R6 here. More precisely, I don't understand why the coeffiecient of $Z$ in $[f_\ast\mathcal{F}]_k$ is equal to $\text{length}_{R_{\mathfrak p}}(M_{\mathfrak p})$. Can you help me?