  The algebra of non-commutative differential geometry (NCG) on the discrete
space $M_4\times Z\ma{N}$ previously proposed by the present author is improved
to give the consistent explanation of the generalized gauge field as the
generalized connection on $M_4\times Z\ma{N}$. The nilpotency of the
generalized exterior derivative ${\mbf d}$ is easily proved. The matrix
formulation where the generalized gauge field is denoted in matrix form is
shown to have the same content with the ordinary formulation using {\mbf d},
which helps us understand the implications of the algebraic rules of NCG on
$M_4\times Z\ma{N}$. The Lagrangian of spontaneously broken gauge theory which
has the extra restriction on the coupling constant of the Higgs potential is
obtained by taking the inner product of the generalized field strength. The
covariant derivative operating on the fermion field determines the parallel
transformation on $M_4\times Z\ma{N}$, which confirms that the Higgs field is
the connection on the discrete space. This implies that the Higgs particle is a
gauge particle on the same footing as the weak bosons. Thus, it is expected
that the mass relation $m\ma{H}=\frac{4}{\sqrt{3}}m\ma{W}\sin\theta\ma{W}$
proposed by the present author holds without any correction in the same way as
the mass relation $m\ma{W}=m\ma{Z}\cos\theta\ma{W}$. The Higgs kinetic and
potential terms are regarded as the curvatures on $M_4\times Z\ma{N}$.
