Absolute Value induces Equivalence not Compatible with Integer Addition

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

Let $\RR$ be the relation on $\Z$ defined as:
 * $\forall x, y \in \Z: \tuple {x, y} \in \RR \iff \size x = \size y$

where $\size x$ denotes the absolute value of $x$.

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

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

However, consider that:

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

while:
 * $1 + 2 = 3$

But it does not hold that:
 * $\size 1 = \size 3$

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

Hence the result, by Proof by Counterexample.