Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema

Theorem
Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Let $h: S \to S$ be a closure operator on $L$.

Then $h$ preserves directed suprema $\struct {h \sqbrk S, \precsim}$ inherits directed suprema.

where
 * $h \sqbrk S$ denotes the image of $h$,
 * $\mathord\precsim = \mathord\preceq \cap \paren {h \sqbrk S \times h \sqbrk S}$

Proof
By Operator Generated by Image of Closure Operator is Closure Operator:
 * $\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} } = h$

where $\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {h \sqbrk S, \precsim}$

Hence the result holds by Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema.