Absolute Value induces Equivalence not Compatible with Integer Addition

Theorem
Let $\Z$ be the set of integers.

Let $\mathcal R$ be the relation on $\Z$ defined as:
 * $\forall x, y \in \Z: \left({x, y}\right) \in \mathcal R \iff \left\vert{x}\right\vert = \left\vert{y}\right\vert$

where $\left\vert{x}\right\vert$ denotes the absolute value of $x$.

Then $\mathcal R$ is not a congruence relation for integer addition.

Proof
From Absolute Value Function on Integers induces Equivalence Relation, $\mathcal R$ is an equivalence relation.

However, consider that:

By conventional integer addition:
 * $-1 + 2 = 1$

while:
 * $1 + 2 = 3$

But it does not hold that:
 * $\left\vert{1}\right\vert = \left\vert{3}\right\vert$

Therefore $\mathcal R$ is not a congruence relation for integer addition.

Hence the result, by Proof by Counterexample.