Dilation pullback: $P_\beta(cX,\bar cY)=k\,P_\alpha$, $|\alpha|=|\beta|=1$, forces $|c|=1$ and $\beta=\alpha$

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Dilation pullback: $P_\beta(cX,\bar cY)=k\,P_\alpha$, $|\alpha|=|\beta|=1$, forces $|c|=1$ and $\beta=\alpha$ · Prove2Me