This particular question is connected (with a slight variation in the definition of $g$) to an earlier question. The link is here. The specifics are:
Given that $u(x,y)$ is the solution of a PDE ($x$ and $y$ are independent variables) we can expand the solution around an arbitrary non-characteristic singularity manifold given by $g(x,y)=0$ in a power series of the form $\displaystyle\sum_{n=0}^{\infty}a_{n}(x,y)g(x,y)^{n+\alpha},$ where $\alpha$ is a negative integer (to be found using leading order analysis).
Given that $g=0$ is assumed to be non-characteristic we have $g_{x}\neq 0$. Thus, by the Implicit Function Theorem we have $g(x,y)=x-f(y)$ near $g=0$, where $f(y)$ is a function of $y$.
For a particular PDE it turns out that $\alpha=-2$ and the coefficients $a_{n}(x,y)$ turn out to be functions of $y$ only.
Given that $g(x,y)=x-f(y)$ why can we replace $a_{n}(x,y)$ by $a_{n}(y)$?