Lemma 45.1.8.label Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exist $p_{1}, p_{2} \in \text{Proj}(M)$ with $p_{1} \sim p_{2}$ and $p_{1} + p_{2} = p$.
Proof, [Lemma 4.1.1, AP]. Let
and $\mathcal{M}\subset \mathcal{F}$ be a maximal family such that $\bracsn{q, r|(q,r) \in \mathcal{M}}$ are all mutually orthogonal. Let $q = \sum_{(q, r) \in \mathcal{M}}q$ and $r = \sum_{(q, r) \in \mathcal{M}}r$, then by Lemma 43.8.10, $q \sim r$, $q \perp r$, and $q + r \le p$.
If $q + r \ne p$, then by Lemma 45.1.7 applied to $p - (q + r)$, there exist non-zero projections $q', r' \in \text{Proj}(M)$ with $q' \sim r'$, $q' \perp r'$, and $q' + r' \le p - (q + r)$, which contradicts the maximality of $\mathcal{M}$.$\square$
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