Reflexive Reduction of Relation Compatible with Group Operation is Compatible/Proof 2

Theorem

Let $\struct {S, \circ}$ be a group.

Let $\RR$ be a relation on $S$ which is compatible with $\circ$.

Let $\RR^\ne$ be the reflexive reduction of $\RR$.

Then $\RR^\ne$ is compatible with $\circ$.

Proof

From the Cancellation Laws, $\circ$ is a cancellable operation.

The result then follows directly from Reflexive Reduction of Relation Compatible with Cancellable Operation is Compatible.

$\blacksquare$