Positive Part of Horizontal Section of Function is Horizontal Section of Positive 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 positive part of $f$.

Proof

Fix $y \in Y$.

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

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

\(=\)

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

\(\ds \)

\(=\)

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

Definition of Positive Part

\(\ds \)

\(=\)

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

Definition of Horizontal Section of Function

\(\ds \)

\(=\)

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

Definition of Positive Part

$\blacksquare$