User:Dfeuer/OR6

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

Suppose that $x \le y$ and $0_R \le z$.

Then $x \circ z \le y \circ z$ and $z \circ x \le z \circ y$.

Proof
We will show that $x \circ z \le y \circ z$; the other conclusion follows from the same argument.

Since $x \le y$, $0_R \le y - x$.

Since $0_R \le y - x$ and $0_R \le z$, Definition:Ordering Compatible with Ring Structure shows that