Inverses of Elements Related by Compatible Relation/Corollary

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

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

Let $x, y \in G$.

Then the following equivalences hold:


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

Proof
Applying User:Dfeuer/CRG2(1) to $x$ and $e$ gives


 * $x \mathrel{\mathcal R} e \iff e \mathrel{\mathcal R} e \circ x^{-1}$

Applying User:Dfeuer/CRG2(3) to $e$ and $x$ gives


 * $e \mathrel{\mathcal R} x \iff e \circ x^{-1} \mathrel{\mathcal R} e$

Since $e \circ x^{-1} = x^{-1}\circ e = x^{-1}$ for all $x \in G$, the theorem holds.