&    \\
    \\
    \quad\quad\quad\ddots &  &   &   &   &   \\
    \\
    & a(\theta_p)\otimes I_{t_p'(x)}  &   &   &    \\
    \\
     &  & f(y_1(x))&  &   \\
   \\
    &  &  &  & \ddots  &  \\
    \\
    &  &   &  &   & f(y_{\bullet}(x))

  \end{array}
  \right)
  \cdot V_xU_x,$$ where $y_1(x),\cdots,y_{\bullet}(x)\in \coprod_{i=1}^{l}[0,1]_i\backslash(W_0^i\cup W_K^i)\subset Sp(A)$ and $U_x,\,t_j(x),\,y_{\bullet}(x)$ may not depend on $x$ continuously.

Define $E_{\widetilde{W_j}}(x)$ follows: let $f_{\widetilde{W_j}}(y)=0_{l_i}$, if $y\in (0,1)_i\backslash(W_0^i\cup W_K^i)$ for each $i\in \{1,2,\cdots,l\}$, and $$a(\theta_s)=
   \begin{cases}
    0_{k_s}, & \mbox{if } s\neq j,  \\
    I_{k_j}, & \mbox{if } s=j.
  \end{cases}$$ Set $$E_{\widetilde{W_j}}(x)=U_x^*V_x^*\cdot
  \left(
  \begin{array}{ccccccc}
   0  &  &  &   &  &  &    \\
    & \ddots &   &   &   &  &   \\
  &   & I_{k_j}\otimes I_{t_j'(x)}  &   &   &   &   \\
   &  &  & \ddots &  &   & \\
  &  &  & & 0&  &    \\
  &  &  &  &  & \ddots  &   \\
  &  & &    &  &   & 0
  \end{array}
  \right)
  \cdot V_xU_x,$$ it does not depend on $x$ continuously, but $E_{\widetilde{W_j}}(x)$ commutes with $\phi_x'(f)$ for all $f\in A$, and if $j_1\neq j_2$, then $E_{\widetilde{W_{j_1}}}(x)E_{\widetilde{W_{j_2}}}(x)=0$.

Suppose that $\{f_{ss'}^j\}\,(1\leq j\leq p,1\leq s,s'\leq k_j)$ are matrix units of $F_1$, for each $t\in[0,1]_i,\,i\in \{1,2,\cdots,l\}$. Let $$g_{ss'}^j(t)=(g_{ss'}^j(t,1),g_{ss'}^j(t,2),\cdots,g_{ss'}^{j}(t,l))$$ where $$g_{ss'}^j(t,i)=
   \begin{cases}
   \varphi_0^i(f_{ss'}^j)\dfrac{\eta-dist(t,[0,\frac{1}{K}]_i)}{\eta}, & \mbox{if } 0\leq t\leq \frac{1}{K}+\eta \\
   0, 