Set Difference of Relations Compatible with Group Operation is Compatible

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

Let $\mathcal R, \mathcal Q$ be relations on $G$ which are compatible with $\circ$.

Then the difference $\mathcal R \setminus \mathcal Q$ is compatible with $\circ$.

Proof
By Complement of Relation Compatible with Group is Compatible, $\complement_{G \times G} \mathcal Q$ is compatible with $\circ$.

Thus by Intersection of Relations Compatible with Operation is Compatible, $\mathcal R \cap \complement_{G \times G}\mathcal Q$ is compatible with $\circ$.

But:
 * $\mathcal R \cap \complement_{G \times G} \mathcal Q = \mathcal R \setminus \mathcal Q$

so the theorem holds.