Horizontal Section of Linear Combination of Functions is Linear Combination of Horizontal Sections

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

Let $f_1, f_2, \ldots, f_n : X \times Y \to \overline \R$ be functions.

Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be real numbers.

Let $y \in Y$.

Then:


 * $\ds \paren {\sum_{k \mathop = 1}^n \alpha_k f_k}^y = \sum_{k \mathop = 1}^n \alpha_k \paren {f_k}^y$

where $f^y$ denotes the $y$-horizontal section of the function $f$.

Proof
Let $x \in X$.

We have:

so:


 * $\ds \paren {\sum_{k \mathop = 1}^n \alpha_k f_k}^y = \sum_{k \mathop = 1}^n \alpha_k \paren {f_k}^y$