Inverses of Elements Related by Compatible Relation

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $\mathcal R$ be a relation compatible with $\circ$.

Let $x, y \in G$.

Then
 * $x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x^{-1}$.

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

By User:Dfeuer/CRG2(1),
 * $x \mathrel {\mathcal R} y \iff e \mathrel {\mathcal R} y \circ x^{-1}$. (1)

By User:Dfeuer/CRG2(2)
 * $y^{-1} \mathrel {\mathcal R} x^{-1} \iff e \mathrel {\mathcal R} \left({y^{-1}}\right)^{-1} \circ x^{-1}$. (2)

Since $\left({y^{-1}}\right)^{-1}=y$ (why?), (2) can be rewritten:
 * $y^{-1} \mathrel {\mathcal R} x^{-1} \iff e \mathrel {\mathcal R} y \circ x^{-1}$. (3)

Now note that the right sides of the double implications in (1) and (3) are identical, so we conclude that
 * $x \mathrel{\mathcal R} y \iff y^{-1} \mathrel{\mathcal R} x$.