Image of Compact Subset under Directed Suprema Preserving Closure Operator

Theorem
Let $L = \struct {S, \preceq}$ be a bounded below algebric lattice.

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

Then:
 * $c \sqbrk {\map K L} = \map K {\struct {c \sqbrk S, \precsim} }$

where
 * $\map K L$ denotes the compact subset of $L$,
 * $c \sqbrk S$ denotes the image of $S$ under $c$,
 * $\mathord \precsim = \mathord \preceq \cap \paren {c \sqbrk S \times c \sqbrk S}$

Proof
We will prove that:
 * $\map K {\struct {c \sqbrk S, \precsim} } \subseteq c \sqbrk {\map K L}$

By Image of Compact Subset under Directed Suprema Preserving Closure Operator is Subset of Compact Subset:
 * $c \sqbrk {\map K L} \subseteq \map K {\struct {c \sqbrk S, \precsim} }$

Thus the result by definition of set equality.