Equivalence Relation is Congruence for Left Operation

Theorem
Every equivalence relation is a congruence for the left operation $\leftarrow$.

Proof
Let $\RR$ be an equivalence relation on the structure $\struct {S, \leftarrow}$.

Then:
 * $x_1 \leftarrow y_1 = x_1$
 * $x_2 \leftarrow y_2 = x_2$

Suppose $x_1 \mathrel \RR x_2 \land y_1 \mathrel \RR y_2$.

It follows directly that:
 * $\paren {x_1 \leftarrow y_1} \mathrel \RR \paren {x_2 \leftarrow y_2}$

Also se

 * Equivalence Relation is Congruence for Right Operation