Inverses of Elements Related by Compatible Relation/Corollary

Corollary to Inverses of Elements Related by Compatible Relation
Let $\struct {G, \circ}$ be a group with identity $e$.

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

Then the following hold:
 * $\forall x, y \in G:$
 * $x \mathrel \RR e \iff e \mathrel \RR x^{-1}$
 * $e \mathrel \RR x \iff x^{-1} \mathrel \RR e$

Proof
From Inverse of Identity Element is Itself:
 * $e^{-1} = e$

From Inverses of Elements Related by Compatible Relation:

Substituting $e$ for $y$ gives:


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

Substituting $e$ for $x$ and $x$ for $y$ gives:


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