This is a question asking for reference.

I have a proof of the following.

Let $f=f(x,y)$ be a smooth function in $\mathbb R^2$ which vanishes at the origin. Then there exist smooth functions $f_1=f_1(xy,x)$ and $f_2=f_2(xy,y)$ such that $f=f_1+f_2$.

Its proof consists on: 1) deal with the formal problem, 2) deal with the flat terms.

However I do not think such a result is new, it must exist somewhere in the literature. Maybe in some more general context. Does anybody knows a reference for such a result?