  A lattice $L$ is said lowly finite if the set $[\mathsf{0},a]$ is finite for
every element $a$ of $L$. We mainly aim to provide a complete proof that, if
$M$ is a subset of a complete lowly finite distributive lattice $L$ containing
its join-irreducible elements, and $a$ an element of $M$ which is not
join-irreducible, then $\displaystyle \sum_{b \in M \cap [\mathsf{0},a]}
\mu_M(b,a)b$ belongs to the submodule $\langle a \wedge b + a \vee b - a - b\
|\ a,b \in L \rangle$ of $\mathbb{Z}L$. That property was originally
established by Zaslavsky for finite distributive lattice. It is essential to
prove the fundamental theorem of dissection theory as will be seen. We finish
with a concrete application of that theorem to face counting for submanifold
arrangements.
