Set of Closed Elements wrt Closure Operator under Subset Operation is Complete Lattice

Theorem
Let $S$ be a set.

Let $\cl$ be a closure operator on the power set $\powerset S$ of $S$.

Let $\mathscr C$ be the set of all subsets $T$ of $S$ such that:
 * $\map \cl T = T$

Then the algebraic structure $\struct {\mathscr C, \subseteq}$ forms a complete lattice.