Image of Compact Subset under Directed Suprema Preserving Closure Operator

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

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

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

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
We will prove that
 * $K \left({\left({c \left[{S}\right], \precsim}\right)}\right) \subseteq c \left[{K \left({L}\right)}\right]$

By Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compzct Subset:
 * $c \left[{K \left({L}\right)}\right] \subseteq K \left({\left({c \left[{S}\right], \precsim}\right)}\right)$

Thus the result by definition of set equality.