Positivity of intersections of two analytic disks at their common boundary point

The goal of this paper is to prove that the index of intersection of two complex curves in a two-dimensional complex manifold touching each other at a common boundary point is positive. This is achieved via the construction of a totally real surface such that the curves in question are attached to it by some parts of their boundaries and then defining a certain ``boundary intersection index'' of two complex ``half-disks'' with their edges on a totally real surface. We prove that this index is always positive. This second result holds true, more generally, for pseudoholomorphic curves with cusps in a two dimensional almost complex manifold, but to our best knowledge is new even for integrable structures, unless the totally real surface in question is supposed to be real analytic.