  We consider classical gauge theory with spontaneous symmetry breaking on a
principal bundle $P\to X$ whose structure group $G$ is reducible to a closed
subgroup $H$, and sections of the quotient bundle $P/H\to X$ are treated as
classical Higgs fields. In this theory, matter fields with an exact symmetry
group $H$ are described by sections of a composite bundle $Y\to P/H\to X$. We
show that their gauge $G$-invariant Lagrangian necessarily factorizes through a
vertical covariant differential on $Y$ defined by a principal connection on an
$H$-principal bundle $P\to P/H$.
