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 $z \circ x < w \circ y$ by transitivity.

Also by Properties of Ordered Ring$(6)$, $z \circ 0 < z \circ x$, so
 * $0 < z \circ x$.