Nonsingular point with $\partial_1 h(z) \neq 0$ forces infinite zero set

Let $h$ be a plane polynomial ($h : \mathrm{PlanePoly}$), $z$ a point of the plane with $h_z : z \in \mathrm{PlaneCurveZeroSet}\,h$, and suppose the partial derivative in the second variable does not vanish at $z$:

Then the zero set of $h$ is infinite:

This is the companion of the $\partial_0$ version, covering nonsingularity detected in the second coordinate. Together the two lemmas show that any finite real algebraic curve has both partials vanishing at every one of its points, i.e. every point is singular; this feeds the singularity-counting bound for irreducible curves used in the Bezout argument.