Restriction of Congruence Relation is Congruence

Definition
Let $\struct {S, \circ}$ be an algebraic structure.

Let $\RR$ be a congruence relation for $\circ$ on $S$.

Let $T \subseteq S$ be a subset of $S$.

Let $\RR {\restriction_T} \subseteq T \times T$ be the restriction of $\RR$ to $T$.

Then $\RR {\restriction_T}$ is a congruence relation for $\circ_T$ on $T$.

Proof
Hence the result by definition of congruence relation.