Lemma 45.1.7.label Let $M$ be a diffuse factor, $p \in \text{Proj}(M)$ be a non-zero projection, then there exist non-zero projections $p_{1}, p_{2} \in \text{Proj}(M)$ with $p_{1} \sim p_{2}$ and $p_{1} + p_{2} \le p$.
Proof, [Lemma 4.1.1, AP]. Since $p$ is not minimal, there exists a projection $q \le p$ with $0 \ne q \ne p$. As $M$ is a factor, $Z(q)Z(p - q) = 1_{M}$. By Proposition 43.8.5, there exist non-zero projections $p_{1}, p_{2} \in \text{Proj}(M)$ with $p_{1} \sim p_{2}$, $p_{1} \le q$, and $p_{2} \le p - q$.$\square$
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