We consider classical gauge theory with spontaneous symmetry breaking on a principal bundle $Pto X$ whose structure group $G$ is reducible to a closed subgroup $H$, and sections of the quotient bundle $P/Hto X$ 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 $Yto P/Hto X$. 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 $Pto P/H$.
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