User:Dfeuer/OR1
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Theorem
Let $\left({R, +, \circ, \le}\right)$ be an ordered ring.
Let $x, y, z \in R$.
Then the following equivalences hold:
- $x \le y \iff x + z \le y + z$
- $x \le y \iff z + x \le z + y$
- $x < y \iff x + z < y + z$
- $x < y \iff z + x < z + y$
Proof
By the definition of an ordered ring, $\left({R, +, \le}\right)$ is an ordered group.
Thus by User:Dfeuer/OG1, the stated equivalences hold.