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 $c\left[{S}\right]$: $x \precsim \sup_P D \implies \exists d \in D: d \precsim x$

Let $D$ be a directed subset of $c\left[{S}\right]$.

By definition of ordered subset:
 * $D$ is directed in $L$.

By definition of algebraic ordered set:
 * $L$ is up-complete.

By definition of up-complete:
 * $D$ admits a supremum in $L$.

By definition of image of set:
 * $\sup_L D \in c\left[{S}\right]$

By Supremum in Ordered Subset:
 * $\sup_L D = \sup_P D$

By definition of ordered subset:
 * $x \preceq \sup_L D$

By definition of closure operator/inflationary:
 * $y \preceq x$

By definition of transitivity:
 * $y \preceq \sup_L D$

By definition of way below relation:
 * $\exists d \in D: y \preceq d$

By definition of subset:
 * $d \in c\left[{S}\right]$

By definition of closure operator/idempotent:
 * $d = c\left({d}\right)$

By definition of closure operator/increasing:
 * $x = c\left({y}\right) \preceq c\left({d}\right) = d$

Thus by definition of ordered subset:
 * $\exists d \in D: x \precsim d$

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:
 * $x \in K\left({P}\right)$