KöMaL 2022 B5229

Let $a\ne 0$ be a real number, and let $f\colon \mathbb{R} \to \mathbb{R}$ be a function, such that $f\big(x+f(y)\big) = f(x) + f(y) + ay$ for all $x,y\in \mathbb{R}$. Prove that $f$ is additive, that is, $f(x+y) = f(x) + f(y)$ for all $x,y\in \mathbb{R}$.