  We construct a gauge field model based on $SU(2)_L\times SU(2)_R\times U(1)_Y
\times \pi_4(SU(2)_L\times SU(2)_R\times U(1)_Y) $ from the principle that both
the original gauge group $G_{YM}$ and the discrete group $\pi_4(G_{YM})$ should
be taken as gauge groups in the sense of non-commutative geometry. We show that
the Yukawa coupling and the Higgs mechanism appear as natural results.
