Equivalence Relation is Congruence for Left Operation

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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:

$\paren {x_1 \leftarrow y_1} \mathrel \RR \paren {x_2 \leftarrow y_2}$

$\blacksquare$


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