I refer systematically to J.L. Kelley's book on General Topology, in particular pages 80-81.
In exercise 2.I, the following theorem is stated:
Theorem. Let $A$ and $B$ be disjoint subsets of a distributive lattice $X$ such that $A$ is an ideal and $B$ is a dual ideal. There there are disjoint sets $A'$ and $B'$ such that $A'$ is an ideal containing $A$, $B'$ is a dual ideal containing $B$, and $A' \cup B'=X$.
In Exercise 2.J The following lemma is stated:
Lemma. If $S$ is a net in $X$, then there is a family $\mathsf{C}$ of subsets of $X$ such that: $S$ is frequently in each member of $\mathsf{C}$, the intersection of two members of $\mathsf{C}$ belongs to $\mathsf{C}$, and for each subset $A$ of $X$ either $A$ of $X \setminus A$ belongs to $\mathsf{C}$.
To prove this Lemma, Kelley suggests to use the previous Theorem, considering $\mathsf{A}$ as the family of all subsets $A$ of $X$ such that $S$ is eventually in $X \setminus A$, and $\mathsf{B}$ as the family of all subsets $B$ of $X$ such that $S$ is eventually in $B$. The ordering $\geq$ is $\subset$.
I must confess that I am unable to follow this route. I see that $\mathsf{A}$ is a dual ideal and that $\mathsf{B}$ is an ideal in $2^X$. Obviously $\mathsf{A}$ and $\mathsf{B}$ are disjoint. The Theorem tells me that there are $\mathsf{A}'$ and $\mathsf{B}'$ which contain $\mathsf{A}$ and $\mathsf{B}$ respectively, with the properties stated in the conclusion. But now, what is $\mathsf{C}$?