Product of Positive Element and Element Greater than One

Theorem
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring with unity $1_R$ and zero $0_R$.

Let $x,y \in R$.

Suppose that $x > 0_R$ and $y > 1_R$.

Then $x \circ y > x$ and $y \circ x > x$.

Proof
A similar argument shows that $y \circ x > x$.