First note that if $g$ solves the equation, then so does $g+cy$ for any constant $c$. Thus, we may reduce to the initial condition $g(0)=-1,g^{\prime}(0)=0$ below. The initial condition specifies a sign choice of the square root, whence

$$\frac{d}{dy}\left(\frac{g}{y}\right)=\frac{1}{y^{2}(1-y^{n})^{\frac{1}{2}}},$$

and the Lemma is reduced to Prop. 5.3. ∎