Geometry of tangent cone to $G$-space of nonpositive curvature with distinguished family of segments

We study a construction of the tangent cone for Busemann $G$-space with distinguished family of segments with additional condition of Busemann curvature nonpositivity. We prove that the constructed cone has geometric properties analogous to the properties of the tangent cone of the standard $G$-space of nonpositive curvature. Earlier the tangent cone construction was used by the first author for proving H. Busemann's conjecture for $G$-spaces of nonpositive curvature stating that every such space is a topological manifold. The constructed tangent cone can be considered as a main tool for the generalization of this theorem to the presented class of spaces.