At a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$, $f\,dt=f\,du$ for some uniformizer $u$ of the local ring

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At a point $(c,d)$ of $w^2=h(t)$ with $h(c)\ne0$, $f\,dt=f\,du$ for some uniformizer $u$ of the local ring · Prove2Me