Equivalence Relation on Integers Modulo 5 induced by Squaring/Multiplication Modulo Beta is Well-Defined

Theorem
Let $\beta$ denote the relation defined on the integers $\Z$ by:
 * $\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$

We have that $\beta$ is an equivalence relation.

Let the $\times_\beta$ operator ("multiplication") on the $\beta$-equivalence classes be defined as:


 * $\eqclass a \beta \times_\beta \eqclass b \beta := \eqclass {a \times b} \beta$

Then such an operator is well-defined.

Proof
That $\beta$ is an equivalence relation is proved in Equivalence Relation on Integers Modulo 5 induced by Squaring.

Let:

We have:

Hence the result.