& \#\big(Sp\phi_{x_2}\cap B(V_r^i,\frac{1}{3}\eta)\big) \\
      &\leq& \#(\Lambda_1({h_r^i})\cup\Lambda_2({h_r^i})) \\
      &\leq& \#\big(Sp\phi_{x_3}\cap B(V_r^i,\frac{2}{3}\eta)\big) \\
      &\leq& \#\big(\Lambda_1({h_r^i})\cup\Lambda_2({h_r^i})\cup\Lambda_3({h_r^i})\big) \\
      &\leq& \#\big(Sp\phi_{x_4}\cap B(V_r^i,\eta)\big)
   \end{aligned}$$

Consider the following two cases:

Case 1.$\quad$ $\Lambda_2({h_r^i})=\Lambda_3({h_r^i})=\varnothing$

In this case, we have $\#(\Lambda_1({h_r^i}))=\#(\Lambda_1({h_r^i})\cup\Lambda_2({h_r^i})\cup\Lambda_3({h_r^i}))$, then $$\#\big(Sp\phi_{x}\cap B(V_r^i,\frac{2}{3}\eta)\big)=\#\big(Sp\phi_x\cap B(V_r^i,\frac{1}{3}\eta)\big).$$ Therefore, $$Sp\phi_x\cap B(V_r^i,\frac{2}{3}\eta)\backslash B(V_r^i,\frac{1}{3}\eta)=\varnothing.$$

Define $\chi_r^i$ as follows: let $a(\theta_j)=0$ for each $j\in \{1,2,\cdots,p\}$, and let $$\chi_r^i(t)=(\chi_r^i(t,1),\chi_r^i(t,2),\cdots,\chi_r^i(t,l)),$$ where $\chi_r^i(t,s)=0_{l_s}$, for $s\neq i$, and $$\chi_r^i(t,i)=
  \begin{cases}
    0_{l_i}, & \mbox{if }\,\, t\in [0,1]_i\backslash\overline{B(V_r^i,\frac{2}{3}\eta)},  \\
     linear, & \mbox{if }\,\, t\in \overline{B(V_r^i,\frac{2}{3}\eta)}\backslash B(V_r^i,\frac{1}{3}\eta),\\
    I_{l_i}, & \mbox{if }\,\, t\in  \overline{B(V_r^i,\frac{1}{3}\eta)}.
  \end{cases}$$ Then $\chi_r^i$ is an element in the center of $A$ and $\phi(\chi_r^i)$ is a projection in $B$. Let $P_r^i=\phi(\chi_r^i)$, then ${\rm rank}\,P_r^i=l_i\#(\Lambda_1({h_r^i}))\geq l_ic_r^i$ and clearly we have $$P_r^i\phi(f)=\phi(\chi_r^i)\phi(f)=\phi(f)\phi(\chi_r^i)=\phi(f)P_r^i,$$ for all $f\in A$.

Case 2.$\quad$ $\Lambda_2({h_r^i})\cup\Lambda_3({h_r^i})\neq\varnothing$

Denote $U_r^i=\overline{B(V_r^i,\eta)}$, then $U_r^i\subset W_r^i$, we have $$\#\big(Sp\phi_{x}\cap U_r^i\big)\geq\#\big(\Lambda_1({h_r^i})\big)+1\geq c_r^i+1.$$ for any $x\in [0,1]\subset Sp(B)$.

We will apply lemma \[2.7\] to $U_r^i$ and any $x'\in [0,1]$, there exists $\eta(x'),\,\delta(x')>0$ small enough(assume that for all $x'\in [0,1],\,\eta(x')\leq \frac{1}{3}\eta$ and $\eta(0)=\eta(1))$ such that $$\#\big (Sp\phi_x\cap B(U_r^i,2\eta(x'))\big)=\#\big (Sp\phi_{x'}\cap U_r^i\big)$$ and $$Sp\phi_x\cap \overline{B(U_r^i,3\eta(x'))}\backslash B(U_r^i,2\eta(x'))=\varnothing$$ for all $x\in B(x',\delta(x'))$.

Define $\chi_r^i$ corresponding to the point $x'\in Sp(B)$ as follows: let $a(\theta_j)=0$ for each $j\in \{1,2,\cdots,p\}$, and let $$\chi_r^i(t)=(\chi_r^i(t,1),\chi_r^i(t,2),\cdots,\chi_r^i(t,l)),$$ where $\chi_r^i(t,s)=0_{l_s}$, if $s\neq i$, and $$\chi_r^i(t,i)=
  \begin{cases}
  0_{l_i}, & 