Properties of Relation Compatible with Group Operation/CRG1

Theorem
Let $\struct {G, \circ}$ be a group.

Let $x, y, z \in G$.

Let $\RR$ be a relation compatible with $\circ$.

Then the following equivalences hold:
 * $x \mathrel \RR y \iff x \circ z \mathrel \RR y \circ z$
 * $x \mathrel \RR y \iff z \circ x \mathrel \RR z \circ y$

Proof
By Relation Compatible with Group Operation is Strongly Compatible, $\RR$ is strongly compatible with $\circ$.

Thus by the definition of strong compatibility, the theorem holds.