Equivalence Relation is Congruence for Constant Operation

Theorem
Every equivalence is a congruence for the constant operation.

Proof
Let $c \in S$.

By the definition of the constant operation, $\forall x, y \in S: x \left[{c}\right] y = c$.

Let $\mathcal R$ be an equivalence relation on $S$.

Every equivalence relation is reflexive, so $c \mathcal R c$.

So:

Hence the result.