Transversality and Strong Tangential Transversality of a Finite Number of Sets

A definition of strong tangential transversality for a finite number of sets is proposed such that strong tangential transversality implies transversality. The main tool in the proofs are infinitesimal characterizations of transversality and subtransversality of a finite number of sets in the prime space which are of independent interest. A normal intersection property for Clarke normal cones, a sum rule for the Clarke subdifferential of a finite number of lower semicontinuous functions, and an abstract Lagrange multiplier rule are obtained as applications.