164 Proof

Suppose that $K=X/Y$. Then, as in frame 161,

$$I+GK=1+\frac{X}{Y}\frac{P}{Q}=\frac{YQ+XP}{YQ}=\frac{1}{YQ};$$

so $\frac{1}{1+GK}=YQ$ and hence $\mathrm{\Phi }$ has all entries in $𝒮$, since