Inverses of Elements Related by Compatible Relation

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

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

Let $x, y \in G$.

Then:
 * $x \mathrel \RR y \iff y^{-1} \mathrel \RR x^{-1}$

Proof
Let $e$ be the group identity of $G$.

By Properties of Relation Compatible with Group Operation/CRG2 $(1)$:


 * $(1): \quad x \mathrel \RR y \iff e \mathrel \RR y \circ x^{-1}$

By Properties of Relation Compatible with Group Operation/CRG2 $(2)$, also:


 * $(2): \quad y^{-1} \mathrel \RR x^{-1} \iff e \mathrel \RR \paren {y^{-1} }^{-1} \circ x^{-1}$

By Inverse of Group Inverse:
 * $\paren {y^{-1} }^{-1} = y$

Thus, we can rewrite $(2)$ as:


 * $(3): \quad y^{-1} \mathrel \RR x^{-1} \iff e \mathrel \RR y \circ x^{-1}$

Now note that the of $(3)$ is the same as the  in $(1)$.

We conclude that:


 * $x \mathrel \RR y \iff y^{-1} \mathrel \RR x$