Multiplying Positive Inequalities

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Theorem

Let $\left({R, +, \circ, \le}\right)$ be an ordered ring.

Let $x, y, z, w \in R$.

Suppose that $0 < x < y$ and $0 < z < w$.


Then $0 < z \circ x < w \circ y$.


Proof

By Properties of Ordered Ring$(6)$:

$z \circ x < z \circ y$
$z \circ y < w \circ y$

Then by transitivity of $\circ$:

$z \circ x < w \circ y$

Also by Properties of Ordered Ring$(6)$:

$z \circ 0 < z \circ x$

Hence by Ring Product with Zero:

$0 < z \circ x$

$\blacksquare$