If $x \in \mathbb{R}$, then we may apply
R480: Basic real addition is commutative to obtain
\begin{equation}
\begin{split}
f_{\mathsf{even}}(-x) = \frac{f(-x) + f(x)}{2} = \frac{f(x) + f(-x)}{2} = f_{\mathsf{even}}(x)
\end{split}
\end{equation}
$\square$
