Completely Normal Space is Normal Space

Theorem
Let $\struct {S, \tau}$ be a completely normal space.

Then $\struct {S, \tau}$ is also a normal space.

Proof
Let $\struct {S, \tau}$ be a completely normal space.

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

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

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

which is precisely the definition of a normal space.