155 Proof

(1) We have shown that $𝐂(s)$ and $𝐂(1/(s+1))$ are isomorphic. The issue is to show that we can choose $P(s)$ and $Q(s)$ coprime in $𝒮$. Since $G(s)$ is rational, we can write $G(s)=M(s)/N(s)$ where complex polynomials $M,N$ have no common zeros.

(2) We introduce a new variable $\lambda =1/(1+s)$ and write

$$\tilde{M}(\lambda )={\lambda }^{m}M\left(\frac{1-\lambda }{\lambda }\right)$$

$$\tilde{N}(\lambda )={\lambda }^{m}N\left(\frac{1-\lambda }{\lambda }\right)$$

where $m$ is the maximum of the degrees of $M$ and $N$, so that $\tilde{M}(\lambda )$ and $\tilde{N}(\lambda )$ are polynomials. Now $\tilde{M}(\lambda )$ and $\tilde{N}(\lambda )$ have no common zeros. The problematic case is $\lambda =0$, but we note that $\tilde{M}(0)$ is the $m^{th}$ coefficient of $M$, and $\tilde{N}(0)$ is the $m^{th}$ coefficient of $N$; so either $\tilde{M}(0)$ or $\tilde{N}(0)$ is not zero by the choice of $m$.