Equivalence Relation induced by Congruence Relation on Quotient Structure is Congruence

Theorem
Let $\struct {A, \circ}$ be an algebraic structure.

Let $\RR$ be a congruence relation on $\struct {A, \circ}$.

Let $\SS$ be a congruence relation on the quotient structure $\struct {A / \RR, \circ_\RR}$ defined by $\RR$.

Let $\TT$ be the relation on $A$ defined as:
 * $\forall x, y \in A: x \mathrel \TT y \iff \eqclass x \RR \mathrel \SS \eqclass y \RR$

Then $\TT$ is a congruence relation on $\struct {A, \circ}$.

Proof
First it is demonstrated that $\TT$ is an equivalence relation.

Let $x_1, y_1, x_2, y_2 \in A$ such that:

Then we have:

Hence the result by definition of congruence relation.