# Strictly Monotone Mapping is Monotone

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## 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 Mapping is Increasing, $\phi$ is increasing.

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

Thus $\phi$ is monotone by definition.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 14$