For each $h\in H(\eta)$, there are mutually orthogonal projections $p_1(h)$, $p_2(h)$, $\cdots$, $p_{m(h)}(h)\in B$ and real numbers $\lambda_1(h),\lambda_2(h),\cdots,\lambda_{m(h)}(h)$ such that $$\|\phi(h)-\sum_{k=1}^{m(h)}\lambda_k(h)p_k(h)\|<\frac{1}{6}.\eqno (*)$$ Denote $$\Lambda(h)=\{\lambda_1(h)^{\thicksim{\rm rank}\,p_1(h)},\cdots,\lambda_{m(h)}(h)^{\thicksim{\rm rank}\,\,p_{m(h)}(h)},
   0^{\sim l'-\sum_{k=1}^{m(h)}{\rm rank}\,p_k(h)}\},$$ $$\Lambda_1(h)=\{\lambda | \lambda\in \Lambda(h),\,\,\lambda\in(1-\frac{1}{6},\,\,1]\},$$ $$\Lambda_2(h)=\{\lambda | \lambda\in \Lambda(h),\,\,\lambda\in(1-\frac{1}{2},1-\frac{1}{6}]\},$$ $$\Lambda_3(h)=\{\lambda | \lambda\in \Lambda(h) ,\,\,\lambda\in(1-\frac{5}{6},1-\frac{1}{2}]\}.$$ Then for any $x\in [0,1]\in Sp(B)$, $Eg(\phi_{x}(h))$ and $\Lambda(h)$ can be paired to within $\frac{1}{6}$ one by one, then $Eg(\phi_{x_0}(h))$ and $Eg(\phi_{x}(h))$ can be paired to within $\frac{1}{3}$ one by one for each $h\in H(\eta)$. We will use these pairing results frequently.

$ Step\,2 $: Set $c_r^i=\#(Sp\phi_{x_0}\cap V_r^i)$, for all $i\in \{1,2,\cdots,l\},\,r\in \{0,1,2,\cdots,K\}$, let us construct $P_r^i$ for each $i\in \{1,2,\cdots,l\},\,r=\{1,2,\cdots,K-1\}$.

For each $x\in [0,1]\subset Sp(B)$, we have $$\phi_x(f,a)=U_x^*\cdot\big( a(\theta_1)^{\sim t_1(x)},\cdots,a(\theta_p)^{\sim t_p(x)},f(y_1(x)),f(y_2(x)),\cdots,f(y_{\bullet}(x))\big)\cdot U_x.$$ Rewrite it by $$\phi_x(f,a)=U_x^*\cdot
  \left(
  \begin{array}{cccccc}
    a(\theta_1)\otimes I_{t_1(x)} &  &  &   &  &    \\
    \\
    \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 U_x,$$ where $y_1(x),\cdots,y_{\bullet}(x)\in \coprod_{i=1}^{l}(0,1)_i\subset Sp(A)$ and $U_x,\,t_j(x),\,y_{\bullet}(x)$ may not depend on $x$ continuously.

For any subset $T\subset (0,1)_i\subset Sp(A)$ and $x\in [0,1]\subset Sp(B)$, define $E_T(x)$ as follows: let $a(\theta_j)=0$ for each 