Separated Sets are Disjoint

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

Let $A, B \subseteq S$ such that $A$ and $B$ are separated in $T$.

Then $A$ and $B$ are disjoint:
 * $A \cap B = \O$

Proof
Let $A$ and $B$ be separated in $T$.

Then: