Vertical/nonvertical irreducible pair meets in at most $d_1 d_2$ points

Let $h$ and $k$ be irreducible bivariate real polynomials ($h_h$, $h_k$) that are not associated ($h_{not}$), of total degree at most $d_1$ and $d_2$ ($h_{deg}$, $k_{deg}$). Suppose $h$ is vertical, $h_{deg0} : (\mathrm{Curry0}\,h).\mathrm{natDegree} = 0$, while $k$ genuinely depends on the first variable, $k_{pos} : 0 < (\mathrm{Curry0}\,k).\mathrm{natDegree}$. Then the intersection is finite with the clean Bezout product bound:

This handles the mixed vertical/nonvertical irreducible case: the vertical curve $h$ contributes at most $d_1$ lines, and on each line $k$ restricts to a nonzero univariate polynomial of degree at most $d_2$. It supplies the sharp $d_1 d_2$ estimate for this case in the case split assembling the full irreducible pair-intersection bound.