Absolute Value is Many-to-One

Theorem
Let $f: \R \to \R$ be the absolute value function:
 * $\forall x \in \R: \map f x = \begin{cases}

x & : x \ge 0 \\ -x & : x < 0 \end{cases}$

Then $f$ is a many-to-one relation.

Proof
Let $\map f {x_1} = y_1, \map f {x_2} = y_2$ where $y_1 \ne y_2$.

The result follows.