Hyperspaces homeomorphic to cones

Let X be a continuum. Suppose that there exists a homeomorphism h :C(X)→cone(Z), where C(X) is the hyperspace of subcontinua of X and Z is a finite-dimensional continuum. In this paper we prove that if Y∈C(X) and h(Y) is the vertex of cone(Z), then Y has the cone=hyperspace property, X−Y has a finite number of components and each one of them is homeomorphic either to [0,∞) or to the real line.