(0,1)_i$, for $i=1,2,\cdots,l$.

Let $$\eta_1=min\{dist(y_r^i,y_s^i),\,y_r^i\neq y_s^i,\, for\ all\ possible \,r,s \, \},$$ $$\eta_2=min\{dist(0_i,V),\,dist(1_i,V),\,dist(y_r^i,V),\,y_r^i\notin V \},$$ $$\eta_3=min\{dist(y_r^i,W_j),\,y_r^i\notin W_j \}.$$ There exists a large enough integer $m$ such that $\frac{1}{m}<\frac{1}{5}min\{\eta_1,\eta_2,\eta_3\}$, set $\eta=\frac{1}{m}$, then we will construct a finite subset $H(\eta)\subset A_+$, by the continuity of $\phi$, there exists $\delta>0$ such that $$\|\phi_{x}(h)-\phi_{x_0}(h)\|<1,$$ for all $h\in H(\eta)$, $x\in B(x_0,\delta)$.

Then by the Weyl spectral variation inequality [@B], $Eg(\phi_x(h))$ and $Eg(\phi_{x_0}(h))$ can be paired to within 1 one by one. By lemma \[2.6\], there exist $X_i(x)\subset Sp\phi_{x}\cap(0,1)_i$, $Y_i(x_0)\subset Sp\phi_{x_0}\cap(0,1)_i$ with $X_i(x)\supset Sp\phi_{x}\cap[\eta,1-\eta]_i$ , $Y_i(x_0)\supset Sp\phi_{x_0}\cap[\eta,1-\eta]_i$ such that $X_i(x)$ and $Y_i(x_0)$ can be paired to within $2\eta$ one by one.

From the construction of $\eta$, we have $V\subset [5\eta,1-5\eta]_i$, and from the pairing results, we have $$\begin{aligned}
      \#\big (Sp\phi_{x_0}\cap V\big) &\leq& \#\big (Sp\phi_x\cap B(V,2\eta)\big) \\
     &\leq & \#\big(Sp\phi_{x}\cap \overline{B(V,3\eta)}\big) \\
     &=& \#\big(Sp\phi_{x_0}\cap \overline{B(V,5\eta)}\big)
  \end{aligned}$$ Similarly, $$\begin{aligned}
      \#_{X_j}\big (Sp\phi_{x_0}\cap W_j\big) &\leq&\#_{X_j}\big (Sp\phi_x\cap B(W_j,2\eta)\big) \\
     &\leq & \#_{X_j}\big(Sp\phi_{x}\cap \overline{B(W_j,3\eta)}\big) \\
     &=& \#_{X_j}\big(Sp\phi_{x_0}\cap \overline{B(W_j,5\eta)}\big)
  \end{aligned}$$ holds for any $x\in [0,1]$.

Since $$\#\big(Sp\phi_{x_0}\cap V\big)=\#\big(Sp\phi_{x_0}\cap \overline{B(V,5\eta)}\big),$$ $$\#_{X_j}\big(Sp\phi_{x_0}\cap W_j\big)=\#_{X_j}\big(Sp\phi_{x_0}\cap \overline{B(W_j,5\eta)}\big).$$

It is obvious that $$\#\big(Sp\phi_x\cap B(V,2\eta)\big)=\#\big(Sp\phi_{x_0}\cap V\big),\,\,
  Sp\phi_x\cap \overline{B(V,3\eta)}\backslash B(V,2\eta)=\varnothing$$ and $$\#_{X_j}\big(Sp\phi_{x}\cap \overline{B(W_j,2\eta)}\big)=\#_{X_j}\big(Sp\phi_{x_0}\cap W_j\big),\,\,
   Sp\phi_x\cap \overline{B(W_j,3\eta)}\backslash B(W_j,2\eta)=\varnothing$$ hold for all $x\in B(x_0,\delta)$.

 Decomposition theorem 
=======================

The following lemma is Proposition 3.2 