Let E be a P. L. space, G = {c^,C2,...,Cm} disjoint, finite subcomplexes, and let GE denote the decomposition space obtained by identifying the Ci to distinct points. The space GE is given a P. L. structure which agrees with that of E outside an open neighborhood of the Ci. If each Ci is full in E and if Ni = N(Ci',E') represents the first derived neighborhood of Ci in E , then GE is shown to be P. L. equivalent to where PiNi represents the cone over the boundary of Ni. If E is a manifold, then the Ni may be taken to be any set of disjoint, regular neighborhoods of the Ci.