We first prove a slightly weaker statement. Namely, say $\dim(X) = N$ and let $X_i \subset X$ be the irreducible components of dimension $N$. Let $\overline{x}_i$ be a geometric generic point of $X_i$. The \'etale local ring $\mathcal{O}_{X, \overline{x}_i}$ is Noetherian of dimension $0$, hence for every coherent $\mathcal{O}_X$-module $\mathcal{F}$ the length $$ m_i(\mathcal{F}) = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{F}_{\overline{x}_i}) $$ is an integer $\geq 0$. We claim that $$ E(\mathcal{F}) = \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) - \sum\nolimits_i m_i(\mathcal{F})\ \chi(Z_i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}|_{Z_i}) $$ is a numerical polynomial in $n_1, \ldots, n_r$ of total degree $< N$. We will prove this using Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}. For any short exact sequence $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ we have $E(\mathcal{F}) = E(\mathcal{F}') + E(\mathcal{F}'')$. This follows from additivity of Euler characteristics (Lemma \ref{lemma-euler-characteristic-additive}) and additivity of lengths (Algebra, Lemma \ref{algebra-lemma-length-additive}). This immediately implies properties (1) and (2) of Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-property-higher-rank-cohomological-variant}. Finally, property (3) holds because for $\mathcal{G} = \mathcal{O}_Z$ for any $Z \subset X$ irreducible reduced closed subspace. Namely, if $Z = Z_{i_0}$ for some $i_0$, then $m_i(\mathcal{G}) = \delta_{i_0i}$ and we conclude $E(\mathcal{G}) = 0$. If $Z \not = Z_i$ for any $i$, then $m_i(\mathcal{G}) = 0$ for all $i$, $\dim(Z) < N$ and we get the result from Lemma \ref{lemma-numerical-polynomial-from-euler}. \medskip\noindent Proof of the statement as in the lemma. Let $Z \subset X$ be the scheme theoretic support of $\mathcal{F}$. Then $\mathcal{F} = i_*\mathcal{G}$ for some coherent $\mathcal{O}_Z$-module $\mathcal{G}$ (Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-coherent-support-closed}) and we have $$ \chi(X, \mathcal{F} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) = \chi(Z, \mathcal{G} \otimes i^*\mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes i^*\mathcal{L}_r^{\otimes n_r}) $$ by the projection formula (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-projection-formula}) and Cohomology of Spaces, Lemma \ref{spaces-cohomology-lemma-relative-affine-cohomology}. Since $|Z| = \text{Supp}(\mathcal{F})$ we see that $Z_i \subset Z$ for all $i$ and we see that these are the irreducible components of $Z$ of dimension $d$. We may and do think of $\overline{x}_i$ as a geometric point of $Z$. The map $i^\sharp : \mathcal{O}_X \to i_*\mathcal{O}_Z$ determines a surjection $$ \mathcal{O}_{X, \overline{x}_i} \to \mathcal{O}_{Z, \overline{x}_i} $$ Via this map we have an isomorphism of modules $\mathcal{G}_{\overline{x}_i} = \mathcal{F}_{\overline{x}_i}$ as $\mathcal{F} = i_*\mathcal{G}$. This implies that $$ m_i = \text{length}_{\mathcal{O}_{X, \overline{x}_i}} (\mathcal{F}_{\overline{x}_i}) = \text{length}_{\mathcal{O}_{Z, \overline{x}_i}} (\mathcal{G}_{\overline{x}_i}) $$ Thus we see that the expression in the lemma is equal to $$ \chi(Z, \mathcal{G} \otimes \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}) - \sum\nolimits_i m_i\ \chi(Z_i, \mathcal{L}_1^{\otimes n_1} \otimes \ldots \otimes \mathcal{L}_r^{\otimes n_r}|_{Z_i}) $$ and the result follows from the discussion in the first paragraph (applied with $Z$ in stead of $X$).