Proposition 7

Let $F$ be a distinguished polynomial and $g, h$ two formal power series of $B$ such that $F = gh$. Then there exists an invertible element $u$ of $B$ such that $ug$ and $u^{-1}h$ are distinguished polynomials and $F = (ug)(u^{-1}h)$.

In fact, the reduced series of $g$ and $h$ are $\neq 0$; hence, by Proposition 6, there exist invertible elements $u, v$ of $B$ such that $ug$ and $vh$ are distinguished polynomials. Then $uvF = (ug)(vh)$ is a distinguished polynomial and $uv$ is invertible. Passing to the reduced series, it is seen immediately that $F$ and $uvF$ have the same reduced order, that is the same degree. The uniqueness assertion in Proposition 6 therefore shows that $F = uvF$, whence $uv = 1$.