External Direct Product of Congruence Relations

Theorem
Let $\struct {A, \odot}$ and $\struct {B, \circledast}$ be algebraic structures.

Let $\RR$ and $\SS$ be congruence relations on $\struct {A, \odot}$ and $\struct {B, \circledast}$ respectively.

Let $\struct {A / \RR, \odot_\RR}$ and $\struct {B / \SS, \circledast_\SS}$ denote the quotient structures defined by $\RR$ and $\SS$ respectively.

Let $\struct {A \times B, \otimes}$ be the external direct product of $\struct {A, \odot}$ and $\struct {B, \circledast}$.

Let $\TT$ be the relation on $\struct {A \times B, \otimes}$ defined as:
 * $\forall \tuple {u, v}, \tuple {x, y} \in A \times B: \tuple {u, v} \mathrel \TT \tuple {x, y} \iff u \mathrel \RR x \land v \mathrel \RR y$

Then:
 * $\TT$ is a congruence relation on $\struct {A \times B, \otimes}$

and the mapping $h: \struct {A / \RR, \odot_\RR} \times \struct {B / \SS, \circledast_\SS} \to \struct {\paren {A \times B} / \TT, \otimes_\TT}$ defined as:
 * $\forall \tuple {\eqclass x \RR, \eqclass y \SS} \in \struct {A / \RR, \odot_\RR} \times \struct {B / \SS, \circledast_\SS}: \map h {\eqclass x \RR, \eqclass y \SS} = \eqclass {\tuple {x, y} } \TT$

is an isomorphism.