Lemma 51.17.4. Let $(A, \mathfrak m)$ be a local ring. If $\mathfrak m = (x_1, \ldots , x_ r)$ and $x_1^{e_1}, \ldots , x_ r^{e_ r}$ are independent for some $e_ i > 0$, then $\text{length}_ A(A/(x_1^{e_1}, \ldots , x_ r^{e_ r})) = e_1\ldots e_ r$.

See [Lech-inequalities] and [Lemma 3 page 300, MatCA].

**Proof.**
Use Lemmas 51.17.2 and 51.17.3 and induction.
$\square$

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