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:
 * $$x \preceq_1 y \implies x = y \or x \prec_1 y$$

So:

$$ $$ $$

This leaves us with:

$$ $$ $$