Ring Product preserves Inequalities on Positive Elements

Theorem

Let $\struct {R, +, \circ, \le}$ be an ordered ring.

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

Let $0 < x < y$ 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:

$0 < z \circ x$

$\blacksquare$