Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not 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 $+_\beta$ operator ("addition") on the $\beta$-equivalence classes be defined as:


 * $\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$

Then such an operation is not well-defined.

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

From Number of Equivalence Classes we have:

We have:

Thus:

Hence the result.