A key point is that there exists a composite bundle P → → P / H → → X where P → → P / H is a principal bundle with the structure group H . Then matter fields, possessing an exact symmetry group H , in the presence of classical Higgs fields are described by sections of some composite bundle E → → P / H → → X , where E → → P / H is some associated bundle to P → → P / H . Herewith, a Lagrangian of these matter fields is gauge invariant only if it factorizes through the vertical covariant differential of some connection on a principal bundle P → → P / H , but not P → → X .