Question 133
A set $A$ of functions $f: \mathbb{R} \rightarrow \mathbb{R}$ is given. For all $f_1, f_2 \in A$ there exists an $f_3 \in A$ such that
$$
f_1\left(f_2(y)-x\right)+2 x=f_3(x+y)
$$
for all $x, y \in \mathbb{R}$. Prove that for all $f \in A$:
$$
f(x-f(x))=0
$$
$$
f_1\left(f_2(y)-x\right)+2 x=f_3(x+y)
$$
for all $x, y \in \mathbb{R}$. Prove that for all $f \in A$:
$$
f(x-f(x))=0
$$