 Solution

Seek $F=L+F_2+F_3+\cdots$, with each $F_d$ homogeneous of degree $d$. Suppose the terms below degree $d$ have been chosen. Comparing degree $d$ ingives a linear equation for $F_d$ whose coefficient is $\pi -{\pi }^d=\pi (1-{\pi }^{d-1})$. The already known term is divisible by $\pi$ because $f(X),g(X)\equiv X^q(\mathrm{mod}\pi )$ and every residue ${\bar{a}}_i\in {\mathbb{F}}_q$ satisfies ${\bar{a}}_i^q={\bar{a}}_i$. Since $1-{\pi }^{d-1}$ is a unit, this determines a unique integral $F_d$. Induction constructs a unique $F\in {\mathcal{O}}_K[[X_1,\dots ,X_n]]$. This is the Lubin–Tate functional equation lemma.