Closure of Complement of Closure is Regular Closed

Theorem
Let $T = \struct {S, \tau}$ be a topological space.

Let $A \subseteq S$ be a subset of $T$.

Let $A^-$ denote the closure of $A$ in $T$.

Let $A'$ denote the complement of $A$ in $S$: $A' = S \setminus A$.

Then $A^{- ' -}$ is regular closed.