Linearly Ordered Space is Completely Normal

Theorem
Let $T = \left({S, \preceq, \tau}\right)$ be a linearly ordered space.

Then $T$ is a completely normal space.

Proof
By Linearly Ordered Space is $T_1$, $T$ is a $T_1$ (Fréchet) space.

By Linearly Ordered Space is $T_5$, $T$ is a $T_5$ space.

Hence the result, by definition of completely normal space.