Completely Normal Space is Normal Space

Theorem
Let $\left({S, \tau}\right)$ be a completely normal space.

Then $\left({S, \tau}\right)$ is also a normal space.

Proof
Let $\left({S, \tau}\right)$ be a completely normal space.

From the definition, $\left({S, \tau}\right)$ is a completely normal space iff:
 * $\left({S, \tau}\right)$ is a $T_5$ space
 * $\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

We have that a $T_5$ space is a $T_4$ space.

So:
 * $\left({S, \tau}\right)$ is a $T_4$ space
 * $\left({S, \tau}\right)$ is a $T_1$ (Fréchet) space.

which is precisely the definition of a normal space.