Diagonal Complement Relation Compatible with Group Operation

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

Let $\Delta$ be the diagonal relation on $G$.

Then $\Delta^c = \complement_{G \times G} \Delta$ is a relation compatible with $\circ$.

In other words, $\ne$ is a relation compatible with $\circ$.

Proof
By Diagonal Relation is Universally Compatible, $\Delta$ is compatible with $\circ$.

By Complement of Relation Compatible with Group is Compatible, $\Delta^c$ is also compatible with $\circ$.