  It is known since the works of Zariski that the essential difficulty in the
local uniformization problem is met already in the case of valuations of height
one. In this paper we prove that local uniformization of schemes and
non-archimedean analytic spaces rigorously follows from the case of valuations
of height one. For non-archimedean spaces this result reduces the problem to
studying local structure of smooth Berkovich spaces.
