  Starting from the well-known quantum Miura transformation for the Lie algebra
$A_n$, we compute explicitly the OPEs for $n=3$ and 4. The primary fields with
spin 3, 4 and 5 are found (for general $n$). By using these primary fields and
the OPEs from quantum Miura transformation, we derive the complete structure of
the nonlinear $W_4$ and $W_5$ algebras.
