# Equivalence Relation is Congruence for Left Operation

## Theorem

Every equivalence is a congruence for the left operation.

## Proof

Let $\mathcal R$ be an equivalence relation on the structure $\left({S, \leftarrow}\right)$.

Then:

$x_1 \leftarrow y_1 = x_1$
$x_2 \leftarrow y_2 = x_2$

Suppose $x_1 \mathop {\mathcal R} x_2 \land y_1 \mathop {\mathcal R} y_2$.

It follows directly that:

$\left({x_1 \leftarrow y_1}\right) \mathop {\mathcal R} \left({x_2 \leftarrow y_2}\right)$

$\blacksquare$