Lubin–Tate functional equation lemma

Given Lubin–Tate series $f,g$ for $\pi$ and a linear form $L=\sum a_i{X}_i$, there is a unique power series $F\equiv L$ modulo terms of degree two satisfyingComparison degree by degree determines each homogeneous term; the Lubin–Tate congruences provide the required divisibility by $\pi$.