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.