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Lemma 10.121.6. Let $R$ be a Noetherian local domain of dimension $1$ with fraction field $K$. Let $V$ be a finite dimensional $K$-vector space. This distance function has the property that

\[ d(M, M'') = d(M, M') + d(M', M'') \]

whenever given three lattices $M$, $M'$, $M''$ of $V$. In particular we have $d(M, M') = - d(M', M)$.

Proof. Omitted. $\square$

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