A Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ carries $V_\alpha$ onto $V_\beta$

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A Möbius map sending $\widehat C_{m,\alpha}$ into $\widehat C_{m,\beta}$ carries $V_\alpha$ onto $V_\beta$ · Prove2Me