Summary: We define the notion of principal fibre bundle that is semi-soldered to the base. Lumiste’s approach to the fundamental group connection on the principal bundle is extended to semi-soldered principal bundles. This leads to a generalization of the notion of Cartan connection. Differential equations for this connection, structure equations for the connection forms and expressions for curvature and torsion objects are found. A tensor of non-degeneration is defined whose vanishing turns the generalized connection into a principal connection. It is proved that the curvature and torsion object of the generalized Cartan connection together with the tensor of non-degeneration is a geometric object.