Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset

Theorem
Let $L = \left({S, \preceq}\right)$ be an algebric lattice.

Let $c:S \to S$ be a closure operator that preserves directed suprema.

Then $c\left[{K\left({L}\right)}\right] \subseteq K\left({\left({c\left[{S}\right], \precsim}\right)}\right)$

where
 * $K\left({L}\right)$ denotes the compact subset of $L$,
 * $c\left[{S}\right]$ denotes the image of $S$ under $c$,
 * $\mathord\precsim = \mathord\preceq \cap \left({c\left[{S}\right] \times c\left[{S}\right]}\right)$

Proof
Let $x \in c\left[{K\left({L}\right)}\right]$.

By definition of image of set:
 * $\exists y \in K\left({L}\right): x = c\left({y}\right)$

and
 * $x \in c\left[{S}\right]$

By definition of compact subset:
 * $y$ is compact in $L$.

By definition of compact element:
 * $y \ll y$

where $\ll$ denotes the way below relation.

Define $P = \left({c \left[{S} \right], \precsim}\right)$ as an ordered subset of $L$.

We will prove that
 * for every directed subset $D$ of $c\left[{S}\right]$: $x \precsim \sup_P D \implies \exists d \in D: d \precsim x$

By definition of way below relation:
 * $x \ll_P x$

By definition:
 * $x$ is a compact element in $P$.

Thus by definition of compact subset:
 * $x \in K\left({P}\right)$