$$\|\phi_x(g)-\widetilde{\phi}_{x}(g)\|<\frac{1}{J}$$ holds for all $x\in [0,1]\subset Sp(B)$, $g\in G$.

Let $g\in G$ be written as $$g=\sum_{i=1}^{l}\sum_{s,s'=1}^{l_i}g_{ss'}^ie_{ss'}^i,$$ where $g_{ss'}^i\in C[0,1]$.

From the definition of $E_{W_r^i}(x)$, we have $$\begin{aligned}
        \widetilde{\phi}_x(g)P_r^i(x)&=& \sum_{iss'}\phi_x(\rho\cdot g_{ss'}^ie_{ss'}^i)(x)\sum_{s=1}^{l_i}\overline{e}_{s1}^i\cdot{\rm new}\,p_r^i(x)\cdot\overline{e}_{1s}^i(x) \\
     &=& \sum_{ss'}\overline{e}_{s1}^i(x)\cdot\phi_x(\rho\cdot g_{ss'}^i)\cdot{\rm new}\,p_r^i(x)\cdot\overline{e}_{1s'}^i(x) \\
     &=& \sum_{ss'}\overline{e}_{s1}^i(x)\cdot{\rm new}\,p_r^i(x)\cdot\widetilde{\phi}_x(\rho\cdot g_{ss'}^i)\cdot\overline{e}_{1s'}^i(x) \\
     &=& \sum_{s=1}^{l_i}\overline{e}_{s1}^i(x)\cdot{\rm new}\,p_r^i(x)\cdot\overline{e}_{1s}^i(x)\sum_{iss'}\widetilde{\phi}_x(\rho\cdot g_{ss'}^ie_{ss'}^i)(x)\\
     &=&  P_r^i(x)\widetilde{\phi}_x(g).
   \end{aligned}$$ Therefore, $$\|P_r^i\phi(g)-\phi(g)P_r^i\|<\frac{2}{J}$$ holds for all $g\in G$.

From the two cases above, we have constructed a collection of mutually orthogonal projections $$P_1^1,\cdots,P_{K-1}^1,\cdots,P_1^l,\cdots,P_{K-1}^l,$$ with ${\rm rank}\,P_r^i=\#\Lambda_1(h_j^i)$ for all $i\in \{1,2,\cdots,l\},\,r\in \{1,2,\cdots,K-1\}$.

$Step\,3$: Now we deal with $V_0^1,V_K^1,\cdots,V_0^l,V_K^l$. Since $$0_i\sim {\rm diag}\{\theta_1^{\thicksim\alpha_{i1}},\theta_2^{\thicksim\alpha_{i2}},\cdots,\theta_p^{\thicksim\alpha_{ip}}\},$$ $$1_i\sim {\rm diag}\{\theta_1^{\thicksim\beta_{i1}},\theta_2^{\thicksim\beta_{i2}},\cdots,\beta_p^{\thicksim\beta_{ip}}\}.$$ $\theta_j$ may appear in different end points, this fact force us to define subsets as follows:

Denote $$\widetilde{V_j}=\bigcup_{\alpha_{ij}\neq0}V_0^i\cup\bigcup_{\beta_{ij}\neq0}V_K^i$$ $$\widetilde{W_j}=\bigcup_{\alpha_{ij}\neq0}W_0^i\cup\bigcup_{\beta_{ij}\neq0}W_K^i$$ for each $j\in \{1,2,\cdots,p\}\,,i\in\{1,2,\cdots,l\}$. Then we turn these $2l$ intervals to $p$ subsets.

Set $X_j=\{\theta_j\}$, we will construct $\widetilde{P_j}$ corresponding to $X_j=\{\theta_j\}$ and $\widetilde{V_j}$, for $j=1,2,\cdots,p$.

Now we will replace the spectra in $\bigcup_{i=1}^lW_0^i$ by $$0_i\sim {\rm diag}\{\theta_1^{\thicksim\alpha_{i1}},\theta_2^{\thicksim\alpha_{i2}},\cdots,\theta_p^{\thicksim\alpha_{ip}}\}$$ and replace the spectra in $\bigcup_{i=1}^lW_K^i$ by $$1_i\sim {\rm diag}\{\theta_1^{\thicksim\beta_{i1}},\theta_2^{\thicksim\beta_{i2}},\cdots,\beta_p^{\thicksim\beta_{ip}}\}$$ to obtain a new homomorphism $\phi'$ from $\phi$ at each point.

Obviously, $$\|\phi_x'(g)-\phi_x(g)\|<\frac{1}{J}$$ holds for all $g\in G$,$x\in [0,1]$.

Then for any given $x\in [0,1]$, there are unitaries $U_x$, $V_x$ such that $$\phi_x'(f,a)=U_x^*V_x^*\cdot
  \left(
  \begin{array}{cccccc}
    a(\theta_1)\otimes I_{t_1'(x)} &  &  &   &  