  This paper provides a non-standard analogue of Bezout's theorem. This is
acheived by showing that, in all characteristics, the notion of Zariski
multiplicity coincides with intersection multiplicity when we consider the full
families of projective degree d and degree e curves. The result is particularly
interesting in that it holds even when we consider intersections at singular
points of curves or when the curves contain non-reduced components. The proof
also provides motivation for the fact that tangency is a definable relation for
families of curves inside a non-linear 1-dimensional Zariski structure X. This
is a crucial ingredient in unpublished work by Peterzil and Zilber that any
such Zariski structure interprets a pure algebraically closed field L with X as
a definable finite cover.
