Points of $w^2=h_\alpha(t)$ with $h_\alpha(c)\ne0$: $t-c$ is a uniformizer and $d(t-c)=dt$

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Points of $w^2=h_\alpha(t)$ with $h_\alpha(c)\ne0$: $t-c$ is a uniformizer and $d(t-c)=dt$ · Prove2Me