Equivalence Relation is Congruence for Right Operation

Theorem
Every equivalence relation is a congruence for the right operation $\rightarrow$.

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

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

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

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

Also see

 * Equivalence Relation is Congruence for Left Operation