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

Theorem
Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

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

Then $h$ preserves directed suprema $\left({h \left[{S}\right], \precsim}\right)$ inherits directed suprema.

where
 * $h \left[{S}\right]$ denotes the image of $h$,
 * $\mathord\precsim = \mathord\preceq \cap \left({h \left[{S}\right] \times h \left[{S}\right]}\right)$

Proof
By Operator Generated by Image of Closure Operator is Closure Operator:
 * $\operatorname{operator}\left({\left({h \left[{S}\right], \precsim}\right)}\right) = h$

where $\operatorname{operator}\left({\left({h \left[{S}\right], \precsim}\right)}\right)$ denotes the operator generated by $\left({h \left[{S}\right], \precsim}\right)$

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