Jet continuation and reducibility of smooth function families

For a family of smooth functions of several variables the problem of local reducibility (through a diffeomorphism) to a lower number of variables is considered. Some sufficient conditions of such reducibility, in terms of subfamily reducibility, are obtained. The argumentation is reduced to solution of a jet continuation problem, for jets defined on the union of a finite family of sets, where the jets can be continued from each of the sets.