Corollary 3.6

Let $L$ be a complete lattice, and let $a\mapsto a^\perp$ satisfy $a\wedge a^\perp=\bot$, $a\vee a^\perp=\top$, both De Morgan identities $(a\wedge b)^\perp=a^\perp\vee b^\perp$ and $(a\vee b)^\perp=a^\perp\wedge b^\perp$, and $(a^\perp)^\perp=a$. Assume also that, for all $a,b,d\in L$, $b\leq a$ and $d\leq a^\perp$ imply $a\wedge(b\vee d)=(a\wedge b)\vee(a\wedge d)$. Let $B\subseteq L$ be nonempty, exclude $\bot$, have a common lower bound in $B$ for every pair of its members, and satisfy $a=(a\wedge b)\vee(a\wedge b^\perp)$ for all $a,b\in B$; assume $B$ is maximal under inclusion among subsets with these properties. Then there exists exactly one subset $S\subseteq L$ containing $B$ such that $S$ contains $\bot$ and $\top$, is closed under $a\mapsto a^\perp$, binary meets, and binary joins, satisfies

for all $a,b,d\in S$, and is maximal under inclusion among subsets with these closure and distributivity properties.