Sum of Positive and Negative Parts

Theorem
Let $X$ be a set, and let $f: X \to \overline{\R}$ be an extended real-valued function.

Let $f^+, f^-: X \to \overline{\R}$ be the positive and negative parts of $f$, respectively.

Then $\left\vert{f}\right\vert = f^+ + f^-$, where $\left\vert{f}\right\vert$ is the absolute value of $f$.