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This article presents the theoretical foundations of a multisensor data fusion system for land vehicle positioning in the framework of the transferable belief model (TBM) applied on reals. Based on this model and using credal inference, a method similar to Fisher's fiducial inference, each source of information produces some belief on the parameter of interest. Sources are data from sensors, to which observation and evolution models are associated. Still sensor failures may occur and the evolution model can be inappropriate in some extremes situations. Conflicts between the individual inferred beliefs are essential data to be taken into account in a multisensor data fusion system. From these conflicts, explicitly given in the TBM, the beliefs produced by each sensor and the a priori information are weighted by discounting coefficients. Furthermore, sensors data are not necessarily precise, they can be imprecise interval-valued. This provides a better representation in case of imprecision. The TBM gives then a frame to deal with imprecision of data modelled in such a way.