Positive Part of Vertical Section of Function is Vertical Section of Positive Part

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

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

Let $x \in X$.

Then:


 * $\paren {f_x}^+ = \paren {f^+}_x$

where:


 * $f_x$ denotes the $x$-vertical function of $f$
 * $f^+$ denotes the positive part of $f$.

Proof
Fix $x \in X$.

Then, we have, for each $y \in Y$: