Reflexive Reduction of Relation Compatible with Group Operation is Compatible

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

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

Let $\prec$ be the reflexive reduction of $\preceq$. That is, $\prec$ is defined as $\preceq \setminus \Delta_S$, where $\Delta_S$ is the diagonal relation on $S$.

Then $\prec$ is compatible with $\circ$.

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

Then by Set Difference of Relations Compatible with Group Operation is Compatible, $\prec$ is compatible with $\circ$.