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.