H(\eta)$, then for each $i\in \{1,2,\cdots,l\}$, then there exists $X_i\subset Sp\phi\cap(0,1)_i$, $Y_i\subset Sp\psi\cap(0,1)_i$ with $X_i\supset Sp\phi\cap[\eta,1-\eta]_i$ , $Y_i\supset Sp\psi\cap[\eta,1-\eta]_i$ such that $X_i$ and $Y_i$ can be paired to within $2\eta$ one by one.

We will apply lemma \[2.5\] to the set $X\triangleq Sp\phi\cap(0,1)_i$, $X_1\triangleq Sp\phi\cap[\eta,1-\eta]_i$  and $Y\triangleq Sp\psi\cap(0,1)_i$, $Y_1\triangleq Sp\psi\cap[\eta,1-\eta]_i$.

For any subset $\widetilde{X}\subset X_1$, let $$X'=\bigcup\{\ [r_s\eta,r_{s+1}\eta]_i\,|\, \widetilde{X}\cap[r_s\eta,r_{s+1}\eta]_i \neq \varnothing \}$$ and $$X''=\{\ x\in [0,1]_i\,|\, dist(x,X')\leq\eta\}.$$ Let $h\in H(\eta)$ corresponding to $X'$, then $h|_{X'}=1$ and $support(h)\subset X''$. Since $Eg(\phi(h))$ and $Eg(\psi(h))$ can be paired within $\varepsilon$, then we will have $$\#\big \{y\in Y,\,dist(y,\widetilde{X})<2\eta\big \}\geq\#(\widetilde{X}).$$ Similarly for any subset $\widetilde{Y}\subset Y_1$,we also have $$\#\big \{x\in X,\,dist(x,\widetilde{Y})<2\eta\big \}\geq\#(\widetilde{Y}).$$ Then there exists $X_0\subset X$ ,$Y_0\subset Y$ with $X_0\supset X_1$, $Y_0\supset Y_1$ such that $X_0$ and $Y_0$ can be paired one by one to within $2\eta$.

\[2.7\] Let $A,B\in \mathcal{C}$, $\phi: A\rightarrow B$ be a unital homomorphism, $V\subset (0,1)_i $ is a closed interval, $W_j$ is a closed neighborhood of $X_j=\{\theta_j\}$, then for any $x_0\in [0,1]$, there exists $\eta,\delta>0$ such that for all $x\in B(x_0,\delta)$,

(1)$\#\big (Sp\phi_x\cap B(V,2\eta)\big)=\#\big (Sp\phi_{x_0}\cap V\big)$,$Sp\phi_x\cap \overline{B(V,3\eta)}\backslash B(V,2\eta)=\varnothing$;

(2)$\#_{X_j}\big(Sp\phi_{x}\cap \overline{B(W_j,2\eta)}\big)=\#_{X_j}\big(Sp\phi_{x_0}\cap W_j\big),\,\,
   Sp\phi_x\cap \overline{B(W_j,3\eta)}\backslash B(W_j,2\eta)=\varnothing$.

Let $\phi_{x_0}:=\pi_{x_0}\circ \phi $. We will denote $Sp\phi_{x_0}$ by $$Sp\phi_{x_0}=\{\theta_1^{\sim t_1(x_0)},\theta_2^{\sim t_2(x_0)},\cdots,\theta_p^{\sim t_p(x_0)},y_1^1,y_2^1,\cdots,y_{\bullet}^1,\cdots,y_1^l,y_2^l,\cdots,y_{\bullet}^l\}$$ where $y_1^i,y_2^i,\cdots,y_{\bullet}^i\in 