Line factor at a root of a vertical curve does not divide a non-associated $k$

Let $h$ and $k$ be irreducible bivariate real polynomials ($h_h$, $h_k$) that are not associated ($h_{not} : \neg\,\mathrm{Associated}\,h\,k$). Suppose $h$ is vertical in the sense $h_{deg0} : (\mathrm{Curry0}\,h).\mathrm{natDegree} = 0$, and let $x : \mathbb{R}$ be a root of its constant curry coefficient, i.e. $h_{xroot} : \mathrm{eval}\,(\_ \mapsto x)\,((\mathrm{Curry0}\,h).\mathrm{coeff}\,0) = 0$. Then the vertical line factor at $x$ does not divide $k$:

This separates the vertical component $h$ from the non-associated curve $k$: even at a root $x$ of $h$'s coefficient, the line $X_1 = x$ is not a component of $k$. It is needed in the mixed vertical/nonvertical intersection case to ensure $k$'s specialization at $x$ is nonzero, so the fiber over $x$ can be bounded by univariate root counting.