Fully Normal Space is Normal Space

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

Then $T$ is a normal space.

Proof
From the definition, $T$ is fully normal :
 * $T$ is fully $T_4$
 * $T$ is a $T_1$ (Fréchet) space.

We have that a fully $T_4$ space is also a $T_4$ space.

So:


 * $T$ is a $T_4$ Space
 * $T$ is a $T_1$ (Fréchet) space

which is precisely the definition of a normal space.