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

Also se

 * Equivalence Relation is Congruence for Right Operation