I have some trouble with proving the following proposition in Handbook of Boolean Algebra by Koppelberg.
Let $A$ be a Boolean algebra and $\mathrm{Ult}A$ the set of ultrafilters in $A.$ The Stone mapping $s$ from $A$ to the powerset algebra $\mathcal{P}(\mathrm{Ult}A)$ is defined by $s(x)=\{p \in \mathrm{Ult}A | x \in p\}.$ Let $M$ be a subset of $A$ such that the supremum $\sum M$ exists and $s$ preserve $\sum M.$(i.e., $s(\sum M) = \cup_{x \in M}s(x)$) Then, there exists a finite subset $M_0$ of $M$ such that $\sum M_0 = \sum M.$
I think $M_0$ should be chosen to satisfy $\cup_{x \in M_0}s(x) = \cup_{x \in M}s(x).$ But I am not familiar with a method to take a finite subset from a subset of a Boolean algebra.
Please give me a hint or advice.