156 Conclusion of proof

(3) By the Lemma, there exist complex polynomials $\tilde{X}(\lambda )$ and $\tilde{Y}(\lambda )$ such that

$$\tilde{M}(\lambda )\tilde{X}(\lambda )+\tilde{N}(\lambda )\tilde{Y}(\lambda )=1.$$

(4) Finally we convert back to the original variable $s=(1-\lambda )/\lambda$ and introduce

$$P(s)=\tilde{M}\left(\frac{1}{1+s}\right);Q(s)=\tilde{N}\left(\frac{1}{1+s}\right);$$

$$X(s)=\tilde{X}\left(\frac{1}{1+s}\right);Y(s)=\tilde{Y}\left(\frac{1}{1+s}\right);$$

so that $P(s),Q(s),X(s),Y(s)$ belong to $𝒮$. Indeed, they are all proper and the only poles are at $s=-1$. Furthermore, $P(s)$ and $Q(s)$ satisfy

$$P(s)X(s)+Q(s)Y(s)=1$$