Gouvêa: proof of the p-adic Weierstrass Preparation Theorem

Question

I am working through Gouvêa's "Introduction to p-adic Numbers". Now I am stuck at Theorem 7.2.6. (p-adic Weierstrass Preparation Theorem). He says that the proof is similar to Proposition 7.2.3. My problem is that the conclusions of these theorems are similar but not the same. For example in the proportion he concludes that for a polynomial $f$ and a polynomial $g$ we have $\lVert g(X)\rVert_1=\lVert f(X)\rVert_1$. But in the theorem he concludes that for a power series $f$ and a polynomial $g$ we have $\lVert f(X)-g(X)\rVert_1<1$. Or in the theorem he claims that we can choose the power series $h$ to start with the term $1$. Where does he get these additional properties about $g$ and $h$ from?