B\u00e9zout bound for a coefficient line against a nonvertical curve

Let $a$ be a nonzero univariate coefficient polynomial of total degree at most $d_1$ and $q$ a nonzero bivariate real polynomial of total degree at most $d_2$, with currying of positive $\mathrm{natDegree}$ ($0 < (\mathrm{Curry}_0(q)).\mathrm{natDegree}$), and assume no root $x$ of $a$ yields a vertical-line factor of $q$ ($\forall x, a(x) = 0 \Rightarrow \neg\,\mathrm{CoeffLineFactor}(x) \mid q$). Then the intersection of the coefficient-line zero set with the plane curve is finite with an explicit product bound:

This handles the mixed vertical/nonvertical intersection case in the B\u00e9zout development: the non-divisibility hypothesis rules out shared vertical components, after which each of the at most $d_1$ vertical lines meets the curve in at most $d_2$ points.