T5 Space is T4 Space

Theorem
Let $\left({X, \vartheta}\right)$ be a $T_5$ space.

Then $\left({X, \vartheta}\right)$ is also a $T_4$ space.

Proof
Let $\left({X, \vartheta}\right)$ be a $T_5$ space.

From the definition of $T_5$ space:

$\left({X, \vartheta}\right)$ is a $T_5$ space iff:
 * $\left({X, \vartheta}\right)$ is a completely normal space
 * $\left({X, \vartheta}\right)$ is a Fréchet ($T_1$) space.

We have that a completely normal space is a normal space.

So:
 * $\left({X, \vartheta}\right)$ is a normal space
 * $\left({X, \vartheta}\right)$ is a Fréchet ($T_1$) space.

which is precisely the definition of a $T_4$ space.