Resultant of coprime primitive curries is nonzero

Let $p$ and $q$ be bivariate real polynomials whose curried forms are primitive ($h_{pprim} : (\mathrm{Curry0}\,p).\mathrm{IsPrimitive}$, $h_{qprim}$ likewise) and coprime ($h_{rel} : \mathrm{IsRelPrime}\,(\mathrm{Curry0}\,p)\,(\mathrm{Curry0}\,q)$). Then the resultant of the curries is a nonzero coefficient polynomial:

This instantiates the abstract fraction-field criterion at the curried polynomials: primitivity plus coprimality over $\mathrm{XCoeff}$ yields coprimality over $\mathrm{XFrac}$ and hence a nonzero resultant. The nonvanishing resultant is what makes the fiber-counting work --- common zeros can only sit above its finitely many roots.