let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$, then the clopen downsets of $X$ are $X_a, a \in L$

Consider a prime ideal $I$ in a clopen downset $D$. The downset generated by $I$ is the intersection of all the sets $X_{a}$ such that $a \notin I$. Because $D$ is compact, there is a finite subintersection which is a subset of $D$. A finite intersection of sets of the form $X_{a}$ is a set of the form $X_{a}$, so for each $I$ in $D$ there is some $a$ such that $I \in X_{a} \subseteq D$. The union of all sets $X_{a} \subseteq D$ is therefore equal to $D$. Compactness yields a finite subunion, and a finite union of sets of the form $X_{a}$ is again a set of the form $X_{a}$.