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 $x \in X$:

\(\ds \map {\paren {f^-}^y} x\)

\(=\)

\(\ds \map {f^-} {x, y}\)

\(\ds \)

\(=\)

\(\ds -\min \set {0, \map f {x, y} }\)

Definition of Negative Part

\(\ds \)

\(=\)

\(\ds -\min \set {0, \map {f^y} x}\)

Definition of Horizontal Section of Function

\(\ds \)

\(=\)

\(\ds \map {\paren {f^y}^-} x\)

Definition of Negative Part

$\blacksquare$