Lemma 55.7.1. Let $n, m_ i, a_{ij}, w_ i, g_ i$ be a numerical type of genus $g$. Given $i, j$ with $a_{ij} > 0$ we have $m_ ia_{ij} \leq m_ j|a_{jj}|$ and $m_ iw_ i \leq m_ j|a_{jj}|$.
Proof. For every index $j$ we have $m_ j a_{jj} + \sum _{i \not= j} m_ ia_{ij} = 0$. Thus if we have an upper bound on $|a_{jj}|$ and $m_ j$, then we also get an upper bound on the nonzero (and hence positive) $a_{ij}$ as well as $m_ i$. Recalling that $w_ i$ divides $a_{ij}$, the reader easily sees the lemma is correct. $\square$
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