Proposition 6 (Preparation Theorem)

Let $f \in B$ be a series whose reduced series is not zero and $s$ its reduced order. Then there exists a unique ordered pair $(u, F)$ such that $u$ is an invertible element of $B$, $F$ a distinguished polynomial of degree $s$ and $f = uF$.

We write $F = X^s + G$, where $G = g_0 + \cdots g_{s-1} X^{s-1}$ ($g_i \in \mathbf{A}$). The relation $f = uF$ is equivalent to $F = u^{-1}f$, that is to $X^s = u^{-1}f - G$. Hence Proposition 5 shows the uniqueness of $G$ and $u^{-1}$ and therefore of $F$ and $u$. It also shows that there exist $v \in B$ and a polynomial $G = g_0 + \cdots + g_{s-1} X^{s-1}$ ($g_i \in \mathbf{A}$) such that $X^s = v - G$; it remains to show that $v$ is invertible in $B$ and that $g_i \in \mathfrak{m}$ for all $i$. Now, writing $\bar{g}_i$ for the canonical image of $g_i$ in $k$ and $\bar{f}, \bar{v}$ for the reduced series of $f, g$,

since $\bar{f}$ is of order $s$, $\bar{g}_i = 0$ for all $i$ and $\bar{v}$ is of order 0, hence $v$ is invertible.