Total Intersection Theorem

If one part of the total intersection group of a curve of order $n$ with a curve of order $n_1+n_2$ constitutes the total intersection with a curve of order $n_1$, then the other part will constitute the total intersection with a curve of order $n_2$.

References

Coolidge, J. L. A Treatise on Algebraic Plane Curves. New York: Dover, p. 32, 1959.