  In this paper we show that the equivalences between certain properties of
closed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]}
hold for closed sets definable in quasianalytic o-minimal structures. In
particular we prove that uniform Chevalley estimate implies a stratification by
the diagram of initial exponents and further, Zariski semicontinuity of the
diagram of initial exponents. We also show that the stratification by the
diagram implies Zariski semicontinuity of Hilbert-Samuel function.
