Relation Compatible with Group Operation is Strongly Compatible/Corollary

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

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

The following equivalences hold:
 * $\forall x, y \in G:$
 * $(1): \quad x \mathrel \RR y \iff e \mathrel \RR y \circ x^{-1}$
 * $(2): \quad x \mathrel \RR y \iff e \mathrel \RR x^{-1} \circ y$


 * $(3): \quad x \mathrel \RR y \iff x \circ y^{-1} \mathrel \RR e$
 * $(4): \quad x \mathrel \RR y \iff y^{-1} \circ x \mathrel \RR e$

Proof
Applying Relation Compatible with Group Operation is Strongly Compatible to $x$, $y$, and $x^{-1}$ we obtain:


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

Applying Relation Compatible with Group Operation is Strongly Compatible to $x$, $y$, and $y^{-1}$, on the other hand, yields:


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

By the definition of inverse element:


 * $x \circ x^{-1} = x^{-1} \circ x = y \circ y^{-1} = y^{-1} \circ y = e$

Making these substitutions proves the theorem.