Image of Directed Suprema Preserving Closure Operator is Algebraic Lattice

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

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

Let $C = \left({c\left[{S}\right], \precsim}\right)$ be an ordered subset of $L$.

Then $C$ is algebraic lattice.

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

By Up-Complete Lower Bounded Join Semilattice is Complete:
 * $L$ is a complete lattice.

By definition of closure operator:
 * $c$ is idempotent.

By Image of Idempotent and Directed Suprema Preserving Mapping is Complete Lattice:
 * $C$ is a complete lattice.

We will prove that
 * $\forall x \in c\left[{S}\right]: x^{\mathrm{compact} }_C$ is directed.