位相幾何学の証明

I am having difficulty with a proof that the Euler characteristic for three-manifolds is 0, i.e. V-E+F-S=0 for a complex K representing the triangulation of a 3-manifold.　The step stating 2E = V1+V2+...Vn eludes me. V1, V2... are the vertices of a boundary complex around a solid sphere surrounding each point Pi in the original complex.
追加。球体の頂点を直線の上にすれば。
We assume that no two spheres around vertices of the 3-manifold complex K overlap. Each edge of K connects two vertices. If we define the vertices of the triangulation of the spheres around vertices Pi to be on edges of K, then each edge defines vertices of two distinct spherical complexes. Then, as V1,V2, ...,Vn are the numbers of vertices of these spheres:
V1+V2+...+Vn = 2E