  We perform an Hamiltonian reduction on a classical \cw(\cg, \ch) algebra, and
prove that we get another \cw(\cg, \ch$'$) algebra, with $\ch\subset\ch'$. In
the case $\cg=S\ell(n)$, the existence of a suitable gauge, called Generalized
Horizontal Gauge, allows to relate in this way two \cw-algebras as soon as
their corresponding \ch-algebras are related by inclusion.
