(F-M) Find the $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$
f(x-f(x-y))+x=f(x+y)
$$
With $x=0$ we write $f(-f(-y))=f(y)$ then $y=0$ gives $f(x-f(x))=f(x)-x$ and $y=x$ gives $f(x-f(0))=f(2x)-x$, but with $y=-x$ we have then

$$
f(x-f(2x))=f(0)-x=f(-f(-(x-f(0))))
$$

and thus $f$ is bijective, so $f(-f(-y))=f(y)$ implies $f=Id$ which is indeed a solution.