c}}{p^{c_ns}} \\
&=& \frac{\|z\|_{\lambda, c}\|h\|_{\lambda, c}^{m+1}}{p^{(m+1)c_ns}}\end{aligned}$$ Now, we know that, $$\begin{aligned}
\left(I + \tau\circ h\right)^{-1}(z) &=& z + \sum_{m=1}^\infty (-1)^{m} (\tau\circ h)^m(z).\end{aligned}$$ Let $w^{(i)}(z) = z + \sum_{m=1}^i (-1)^{m} (\tau\circ h)^m(z)$. We claim that the sequence $\left(w^{(i)}(z)\right)_{i=1}^\infty$ is Cauchy for every $z$ in $F[X; Y, \lambda]_c$. Indeed, $$\begin{aligned}
 \|w^{(i+1)}(z) - w^{(i)}(z)\|_{\lambda, c} &=& \|(\tau\circ h)^{i+1}(z)\|_{\lambda, c} \\
&\leq& \frac{\|z\|_{\lambda, c}\|h\|_{\lambda, c}^{i+1}}{p^{(i+1)c_ns}}.\end{aligned}$$ Since $\|h\|_{\lambda, c} < p^{c_ns}$, we see that this difference approaches $0$ as $i$ approaches $\infty$. Since this norm is non-Archimedean, this is all we need to show. Therefore, since $F[X; Y, \lambda]_c$ is complete, we see that $w(z) = \lim_i w^{(i)}(z)$ exists for every $z$ in $F[X; Y, \lambda]_c$. Uniqueness is immediate from the invertibility of the map.

To prove the last statement, note that we could already carry out division uniquely in the ring $F[X_1, \cdots, X_{n-1}; Y,
\lambda]_c[X_n]$, by the polynomial Euclidean algorithm. Therefore, the division is unique in $F[X; Y, \lambda]_c$.

Suppose that $g$ is $X_n$-distinguished of degree $s$ in $F[X; Y, \lambda]$, and that $f$ is an element of $F[X; Y, \lambda]$. Then there exist unique elements, $q$ in $F[X; Y, \lambda]$, and $r$ in the polynomial ring $F[X_1, \cdots, X_{n-1}; Y, \lambda][X_n]$ with 