Lemma 10.59.9. Let $R$ be a Noetherian local ring. Let $I \subset R$ be an ideal of definition. Let $M$ be a finite $R$-module which does not have finite length. If $M' \subset M$ is a submodule with finite colength, then $\chi _{I, M} - \chi _{I, M'}$ is a polynomial of degree $<$ degree of either polynomial.

**Proof.**
Follows from Lemma 10.59.2 by elementary calculus.
$\square$

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