This is a follow-up question on [this](https://mathoverflow.net/questions/229185/collection-of-projection-operators-in-finite-dimension-and-algebraic-techinques/229295#229295). Let $A$ be a von Neumann algebra and $P$ be its projection lattice. 


For $p,s,q \in P$, let us define $ p \perp q \mid s \iff ps^\perp q = 0$ where $s^\perp = 1-s$.


Let $P\_0 \subset P$ be a complete sublattice of $P$. Is the following true over $P\_0$ (i.e., for any $p,q,r,s \in P\_0$):
\begin{align\*}
 r \perp p \mid (q \vee s), \quad r \perp q \mid (p \vee s) \implies r \perp (p\vee q) \mid s\;?
\end{align\*}



A relation among projections of a von Neumann algebra
I think it is false - let $r=1$ and take any distinct rank $1$ projections $p,q,s\in M\_2$ so $p\vee q=p\vee s=q\vee s=1$.


