we have defined $\widetilde{\pi}_e^{j'}:B\rightarrow  M_{k_{j'}'}(\mathbb{C})$, $\widetilde{\pi}_0^{j'}: \varphi_0'(F_1')\rightarrow M_{k_{j'}'}(\mathbb{C})$, $\widetilde{\pi}_1^{j'}: \varphi_1'(F_1')\rightarrow M_{k_{j'}'}(\mathbb{C})$, as follows: $$\widetilde{\pi}_e^{j'}:B\rightarrow  M_{k_{j'}'}(\mathbb{C}),$$ $$\widetilde{\pi}_0^{j'}\circ\varphi_0'(a')={\rm sgn}(\alpha_{j'}')\cdot a'(\theta_{j'}'),$$ $$\widetilde{\pi}_1^{j'}\circ\varphi_1'(a')={\rm sgn}(\beta_{j'}')\cdot a'(\theta_{j'}'),$$ where sgn$(x)$ is the sign function.

Since $\overline{p}_r^i(0),\,{q_r^i}|_{O_0}(0)\in \varphi_0'(F_1'),\,\,
   \overline{p}_r^i(1),\,{q_r^i}|_{O_1}(1)\in \varphi_1'(F_1')$, then

$${\rm rank}\,\,\widetilde{\pi}_0^{j'}(\overline{p}_r^i(0))\leq \#(Sp(\widetilde{\pi}_0^{j'}\circ\pi_0\circ\phi)\cap U_r^i)
   \leq{\rm rank}\,\,\widetilde{\pi}_0^{j'}({q_r^i}|_{O_0}(0))$$ $${\rm rank}\,\,\widetilde{\pi}_1^{j'}(\overline{p}_r^i(1))\leq \#(Sp(\widetilde{\pi}_1^{j'}\circ\pi_1\circ\phi)\cap U_r^i)
   \leq{\rm rank}\,\,\widetilde{\pi}_1^{j'}({q_r^i}|_{O_1}(1))$$ holds for each $j'\in \{1,2,\cdots,p'\}$ and if $j'$ satisfies $\alpha_{j'}'>0,\,\beta_{j'}'>0$, then $$\widetilde{\pi}_0^{j'}(\overline{p}_r^i(0))=\widetilde{\pi}_1^{j'}(\overline{p}_r^i(1))=\pi_e^{j'}(\overline{p}_r^i)
   \in M_{k_{j'}'}(\mathbb{C})$$ and $$\widetilde{\pi}_0^{j'}({q_r^i}|_{O_0}(0))=\widetilde{\pi}_1^{j'}({q_r^i}|_{O_1}(1))=\pi_e^{j'}\circ\phi(\widetilde{\chi}_r^i)\in M_{k_{j'}'}(\mathbb{C}).$$

Then there exists a collection of unitaries $u_{j'}\in M_{k_{j'}'}(\mathbb{C})$, such that $$u_{j'}^*\cdot\widetilde{\pi}_0^{j'}(\overline{p}_r^i(0))\cdot u_{j'}<\widetilde{\pi}_0^{j'}({q_r^i}|_{O_0}(0)),$$ $$u_{j'}^*\cdot\widetilde{\pi}_0^{j'}(\overline{p}_r^i(0))\cdot u_{j'}<\widetilde{\pi}_1^{j'}({q_r^i}|_{O_1}(1)).$$ Hence, $$\varphi_0'(u_1^*,u_2^*,\cdots,u_{p'}^*)\cdot\overline{p}_r^i(0)\cdot\varphi_0'(u_1,u_2,\cdots,u_{p'})<{q_r^i}|_{O_0}(0),$$ $$\varphi_1'(u_1^*,u_2^*,\cdots,u_{p'}^*)\cdot\overline{p}_r^i(1)\cdot\varphi_1'(u_1,u_2,\cdots,u_{p'})<{q_r^i}|_{O_1}(1).$$ Connect $\varphi_0'(u_1,u_2,\cdots,u_{p'})$ and $\varphi_1'(u_1,u_2,\cdots,u_{p'})$ by a unitary path $v(t)\in B$, then $v^*\cdot\overline{p}_r^i\cdot v$ belongs to $B$.

Since $$p_r^i(x)<{q_r^i}|_{O_0}(x),\,\,\forall x\in O_0,$$ $$p_r^i(x)<{q_r^i}|_{O_1}(x),\,\,\forall x\in O_1,$$ fix $\epsilon'>0$ small enough, then we can connect $v^*(0)\cdot\overline{p}_r^i(0)\cdot v(0)$ and $p_r^i(\epsilon')$ by a path new $p_r^i(x),\,0\leq x\leq\epsilon'\leq\delta(0)$ with the property that new $p_r^i(x)<{q_r^i}|_{O_0}(x)$. A similar construction can be carried out in $1-\delta(1)\leq 1-\epsilon'\leq x\leq 1$. This will ensure that new $p_r^i$ belongs to $B$.

Set $$P_r^i(x)=\sum_{s=1}^{l_i}\overline{e}_{s1}^i(x)\cdot{\rm new}\,p_r^i(x)\cdot\overline{e}_{1s}^i(x);$$ Now we have $P_r^i(x)\leq E_{W_r^i}(x)$.

Choose arbitrary $z_r^i\in W_r^i$, if we change all the spectra in $Sp(\phi_x)\cap W_r^i$ to $z_r^i$, we obtain a pointwise homomorphism $\widetilde{\phi}_x$.

Since Diameter $(W_r^i)<\frac{2}{K}$ and for any $x,x'\in [0,1]_i \subset Sp(A)$ with $d(x,x')<\frac{2}{K}$, $\|g(x)-g(x')\| <\frac{1}{J}$ holds for all $g\in G$. Then we have that 