Silverman and Tate in Rational Points On Elliptic Curves (and Tao here) discuss a proof of the Cayley-Bacharach Theorem in the case where two conics intersect at nine distinct points. The proof uses Bézout's theorem, but other than that is elementary.
Is there a way to modify the aforementioned proof (or, present some other simple proof) that can prove Caley-Bacharach in the case where the nine intersection points of the conics are not necessarily distinct (that is, the number of intersection points counting multiplicity is nine, but not counting multiplicity is less than nine)?