Strictly Monotone Mapping is Monotone

Theorem
A mapping that is strictly monotone is a monotone mapping.

Proof
If $\phi$ is strictly monotone, then it is either strictly increasing or strictly decreasing.

If $\phi$ is strictly increasing, then by Strictly Increasing is Increasing, $\phi$ is increasing.

If $\phi$ is strictly decreasing, then by Strictly Decreasing is Decreasing, $\phi$ is decreasing.

Thus $\phi$ is monotone, from the definition of monotone.