  We prove that the sequence of MacLane key polynomials constructed in
\cite{Mac1} and \cite{Sp2} for a valuation extension $(K,\nu)\subset
(K(x),\mu)$ is finite, provided that both $\nu$ and $\mu$ are divisorial and
$\mu$ is centered over an analytically irreducible local domain
$(R,\frak{m})\subset K[x]$. As a corollary, we prove Izumi's theorem on
comparison of divisorial valuations. %We show that the existence of Izumi
constants is equivalent to the finiteness of the sequence of the MacLane
key-polynomials. We give explicit bounds for the Izumi constant in terms of the
key polynomials of the valuations. We show that this bound can be attained in
some cases.
