Inverses of Elements Related by Compatible Relation/Corollary

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

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

Let $x, y \in G$.

Then the following equivalences hold:
 * $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
 * $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$

Proof
Applying Properties of Relation Compatible with Group Operation/CRG2$(1)$ to $x$ and $e$ gives:


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

Applying Properties of Relation Compatible with Group Operation/CRG2$(3)$ to $e$ and $x$ gives:


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

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