Negative Part of Horizontal Section of Function is Horizontal Section of Negative Part

Theorem
Let $X$ and $Y$ be sets.

Let $f : X \times Y \to \overline \R$ be a function.

Let $y \in Y$.

Then:


 * $\paren {f^y}^- = \paren {f^-}^y$

where:


 * $f^y$ denotes the $y$-horizontal function of $f$
 * $f^-$ denotes the negative part of $f$.

Proof
Fix $y \in Y$.

Then, we have, for each $x \in X$: