Strictly Increasing Mapping is Increasing

Theorem
A mapping that is strictly increasing is an increasing mapping.

Proof
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be strictly increasing.

Note that, from Strictly Precedes is a Strict Ordering:
 * $x \ \preceq_1 \ y \implies x = y \lor x \ \prec_1 \ y$

So:

This leaves us with: