\mbox{if }\,\, t\in [0,1]_i\backslash\overline{B(U_r^i,3\eta(x'))},  \\
  linear, & \mbox{if }\,\,t\in \overline{B(U_r^i,3\eta(x'))}\backslash B(U_r^i,2\eta(x')),\\
   e_{11}^i & \mbox{if }\,\, t\in \overline{B(U_r^i,2\eta(x'))}.
  \end{cases}$$ here we use $e_{11}^i$ as the matrix units of $M_{l_i}(\mathbb{C})$.

Then ${\chi_r^i}\in A$ and $\phi({\chi_r^i})$ defines a continuous projection-valued function on a certain open neighbourhood $B(x',\delta(x'))$ and denote it by ${q_r^i}|_{B(x',\delta(x'))}(x)$, then $${\rm rank}\,{q_r^i}|_{B(x',\delta(x'))}(x)\geq \#\big (Sp\phi_x\cap B(U_r^i,\eta(x'))\big)\geq \#\big(Sp\phi_{x}\cap U_r^i\big)\geq \#\big(\Lambda_1({h_r^i})\big)+1$$ for any $x\in B(x',\delta(x'))$.

Let $\mathcal{W}=\{B(x',\delta(x'),x'\in [0,1]\}$, then $\mathcal{W}$ is an open cover of $[0,1]$, apply lemma \[3.1\] (selection principle), there exists a projection-valued function $p_r^i$ defined on the whole set $[0,1]$ with ${\rm rank}\ p_r^i=\#(\Lambda_1(h_r^i))$, and $$p_r^i(x)\leq\bigvee\{{q_r^i}|_{B(x',\delta(x'))}(x)\,\,|B(x',\delta(x')\in\mathcal{W}\}\leq\overline{e}_{11}^i(x).$$

In general, $p_r^i(x)$ belongs to $C([0,1],M_{l'}(\mathbb{C}))$, not to $B$, and therefore, we need to make suitable changes near the endpoints. We require that 0 and 1 each belong to only one of the open sets in the cover in $\mathcal{W}$, say $O_0(=B(0,\delta(0)))$ and $O_1(=B(0,\delta(1)))$, respectively. Since we assume that $\eta(0)=\eta(1)$, then the elements $\chi_r^i$ corresponding to 0 and $\chi_r^i$ corresponding to 1 are the same, denote this element by $\widetilde{\chi}_r^i$ and we have $${q_r^i}|_{O_0}(0)=\pi_0\circ\phi({\widetilde{\chi}_r^i})\,\,\,{\rm and}\,\,\,\pi_1\circ\phi({\widetilde{\chi}_r^i})={q_r^i}|_{O_1}(1).$$

Denote $$\overline{p}_r^i=\sum_{\lambda_k(h_r^i)\in \Lambda_1(h_r^i)}\,p_k(h_r^i).$$ Then $\overline{p}_r^i\in B$ (see notation in ($*$) and the paragraph below), and $${\rm  rank}\,\,\overline{p}_r^i = {\rm rank}\,\, p_r^i=\#(\Lambda_1(h_r^i)).$$ Recall that for each $j'\in\{1,2,\cdots,p'\}$ and $a'=(a'(\theta_1'),a'(\theta_2'),\cdots,a'(\theta_{p'}'))\in F_1'$, 