# Equivalence Relation is Congruence for Constant Operation

## Theorem

Every equivalence relation is a congruence relation for the constant operation.

## Proof

Let $c \in S$.

By the definition of the constant operation:

$\forall x, y \in S: x \sqbrk c y = c$

Let $\RR$ be an equivalence relation on $S$.

Every equivalence relation is reflexive, so:

$c \mathrel \RR c$

So:

 $\displaystyle x_1 \mathrel \RR x_2$ $\land$ $\displaystyle y_1 \mathrel \RR y_2$ $\displaystyle \leadsto \ \$ $\displaystyle \paren {x_1 \sqbrk c y_1}$ $\RR$ $\displaystyle \paren {x_2 \sqbrk c y_2}$ True Statement is implied by Every Statement

Hence the result.

$\blacksquare$