Lemma 55.3.10. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type. Let $e$ be the number of pairs $(i, j)$ with $i < j$ and $a_{ij} > 0$. Then the expression $g_{top} = 1 - n + e$ is $\geq 0$.

**Proof.**
If not, then $e < n - 1$ which means there exists an $i$ such that $a_{ij} = 0$ for all $j \not= i$. This contradicts assumption (3) of Definition 55.3.1.
$\square$

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