  We consider classical gauge theory on a principal bundle P->X in a case of
spontaneous symmetry breaking characterized by the reduction of a structure
group G of P->X to its closed subgroup H. This reduction is ensured by the
existence of global sections of the quotient bundle P/H->X treated as classical
Higgs fields. Matter fields with an exact symmetry group H in such gauge theory
are considered in the pairs with Higgs fields, and they are represented by
sections of a composite bundle Y->P/H->X, where Y->P/H is a fiber bundle
associated to a principal bundle P->P/H with a structure group H. A key point
is that a composite bundle Y->X is proved to be associated to a principal
G-bundle P->X. Therefore, though matter fields possess an exact symmetry group
H, their gauge G-invariant theory in the presence of Higgs fields can be
developed. Its gauge invariant Lagrangian factorizes through the vertical
covariant differential determined by a connection on a principal H-bundle
P->P/H. In a case of the Cartan decomposition of a Lie algebra of G, this
connection can be expressed in terms of a connection on a principal bundle
P->X, i.e., gauge potentials for a group of broken symmetries G.
