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Let k be a field of any characteristic and not necessarily algebraically closed. In this paper we extend some results of the transversality theory to larger families of k-schemes and of k-morphisms and to all the local geometric properties satisfying some given axioms (for example: geom. regular, geom. reduced, geom. normal and the properties Sr and Rs). As an application, we obtain a generalization of the well known Bertini type theorems for local geometric axiomatic properties.