Ring Product preserves Inequalities on Positive Elements

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:


 * $0 < z \circ x$