A torse-forming bundle of positions was defined and it was shown that if the space permits such a bundle, conditions are satisfied in the adapted coordinate system (2.6). The basic theorems are the following: Theorem 1. The Riemannian space $V_n$ which is the composition of $x_qXx_{n-q}$ is a semi-decomposed Riemannian space if the following conditions are satisfied: I) The bundle of positions of the base space $x_q$ is Chebishevian; IIa) On the base space $x_q$ the affine connection is given, while the connections of a analogous positions are in projective relationship; or: IIb) On the base space $x_q$ the Riemannian metric is given, while the metrics of analogous positions are in conformal correspondence; III) The bundle of positions of the base space $x_{n-q}$ is torse-forming; IV) The transversal positions are orthogonal. Theorem 2. The Riemannian space $V_n$ which is the composition of $xqXx_{n-2q}X\bar x$ is a Riemannian extension of the $q$-dimensional space of the affine connection (or of the $q$-dimensional conformal space) if the following conditions are satisfied: I') The bundle of positions of she base space $x_{n-2q}$ is Chebishevian in relation to the positions of the base space $x_q$; IIa') On the base $x_q$ the affine connection is given, while the connections of analogous positions are in projective correspondence; or; IIb') On the base space $x_q$ the Riemannian metric is given, while the metrics of analogous positions are in conformal correspondence; III') The bundle of positions of the base space $x_q$ is torse-forming; IV') The positions of the base space $x_q$ are totally isotropic manifolds.