{\rm sgn}(\beta_{ij})\cdot\varphi_1^i(a), & \mbox{if }\,\, t\in \overline{B(V_K^i,\frac{1}{3}\eta)}.
    \end{cases}$$ Then $\chi_j\in A$ and $\phi(\chi_j)$ is a projection in $B$. Set $P_j=\phi(\chi_j)$, then $P_j$ commutes with $\phi(f)$ for all $f\in A$ and $${\rm rank}\,P_j=k_j\#(\Lambda_1({h_j}))\geq k_jt_j+k_j\sum_{i=1}^{l}\alpha_{ij}c_0^i+k_j\sum_{i=1}^{l}\beta_{ij}c_k^i.$$

Case 2.$\quad$ $\Lambda_2({h_j})\cup\Lambda_3({h_j})\neq\varnothing$

Denote $U_j\triangleq \overline{B(\widetilde{V_j},\eta)}$, then we have $$\#\big(Sp\phi_{x}\cap U_j\big)\geq\#\big(\Lambda_1({h_j})\big)+1\geq t_i+\sum_{i=1}^{l}\alpha_{ij}c_0^i+\sum_{i=1}^{l}\beta_{ij}c_k^i+1.$$ for any $x\in [0,1]$.

Apply lemma \[2.7\] to $U_j$ and any $x'\in [0,1]$, we may require that $\eta'(x'),\delta'(x')$ small enough (we still assume that for all $x'\in [0,1],\,\eta'(x')\leq \frac{1}{3}\eta$ and $\eta'(0)=\eta'(1))$ and also satisfy $$\#_{X_j}\big(Sp\phi_{x}\cap \overline{B(U_j,2\eta'(x'))}\big)=\#_{X_j}\big(Sp\phi_{x_0}\cap U_j\big),\,\,
   Sp\phi_x\cap \overline{B(U_j,3\eta'(x'))}\backslash B(U_j,2\eta'(x'))=\varnothing$$ for all $x\in B(x',\delta'(x'))$.

Define $\chi_j$ corresponding to the point $x'$ as follows: Set $$\chi_j(t)=(\chi_j(t,1),\chi_j(t,2),\cdots,\chi_j(t,l)),$$ where $$\chi_j(t,i)=
   \begin{cases}
    {\rm sgn}(\alpha_{ij})\cdot\varphi_0^i(f_{11}^j),\,\, & \mbox{if }\,\,t\in [0,\frac{1}{K}]_i{\cap}\overline{B(U_j,2\eta'(x'))},  \\
    linear, & \mbox{if }\,\, t\in [0,\frac{1}{K}]_i{\cap}\overline{B(U_j,3\eta'(x'))}\backslash B(U_j,2\eta'(x')),\\
    0_{l_i}, & \mbox{if }\,\, t\in [0,1]_i\backslash B(U_j,3\eta'(x')), \\
    linear, & \mbox{if }\,\, t\in [1-\frac{1}{K},1]_i{\cap}\overline{B(U_j,3\eta'(x'))}\backslash B(U_j,2\eta'(x')),\\
    {\rm sgn}(\beta_{ij})\cdot\varphi_1^i(f_{11}^j), & \mbox{if }\,\, t\in [1-\frac{1}{K},1]_i{\cap}\overline{B(U_j,2\eta'(x'))}.
    \end{cases}$$ Then $\chi_j\in A$ and $\phi({\chi_j})$ defines a continuous projection-valued function on an open neighbourhood $B(x',\delta'(x'))$ and denote it by ${q_j}|_{B(x',\delta'(x'))}(x)$, then $${\rm rank}\,\,{q_j}|_{B(x',\delta'(x'))}(x)\geq \#_{X_j}\big (Sp\phi_x\cap B(U_j,\eta'(x'))\big)\geq \#_{X_j}\big(Sp\phi_{x}\cap U_j\big)$$ for each $x\in B(x',\delta'(x'))$.

Apply lemma \[3.1\], there is a projection-valued function $p_j$ defined on the whole set $[0,1]$ with ${\rm rank}\ p_j=\#(\Lambda_1({h_j}))$, and $$p_j(x)\leq\bigvee\{{q_j}|_{B(x',\delta'(x'))}(x)\,|\,B(x',\delta'(x')\in\mathcal{W}\}\leq\overline{f}_{11}^j(x).$$

In general, $p_j(x)$ does not belong to 