A\rightarrow B$ satisfies $\phi(H(\eta))\subset_{\frac{1}{6}}\{ f\in B\, |f\,has\,finite\,spectrum\}$, then there exists a projection $q\in B$ and a unital homomorphism $\psi: A\rightarrow (1-q)B(1-q)$ with finite dimensional image such that

$(1)\,\,L\cdot{\rm rank}\,\,(\phi(e)-\psi(e))<{\rm rank}\,\,\psi(e)$ for any projection $e\in A$,

$(2)\,\,\|q\phi(g)-\phi(g)q\|<\dfrac{4}{J} \,\,for\, \,any\,\, g\in G$,

$(3)\,\,\|\phi(g)-q\phi(g)q\oplus\psi(g)\|<\dfrac{4}{J} \,\,for\,\,any\,\, g\in G$.

Since $G\subset A$ is a finite set, $K_0(A)$ is finitely generated, we may assume that $G$ contains a set of projections which generate $K_0(A)$. Then there exists an integer $K>0$ such that for any $x,x'\in [0,1]_i$ with $d(x,x')<\frac{2}{K}$, $\|g(x)-g(x')\| <\frac{1}{J}$ holds for all $g\in G$. Set $\eta=\frac{1}{8K(L+1)}$ and take $H(\eta)$ as test functions.

Let $A=A(F_1,F_2,\varphi_0,\varphi_1)$, $B=B(F_1',F_2',\varphi_0',\varphi_1')$, where $$F_1=\bigoplus_{j=1}^p M_{k_j}(\mathbb{C}),\,\,F_2=\bigoplus_{i=1}^l M_{l_i}(\mathbb{C}),\,\,
  F_1'=\bigoplus_{j'=1}^{p'}M_{k_{j'}'}(\mathbb{C}),\,\,F_2'=M_{l'}(\mathbb{C}).$$ Let $\varphi_{0*},\varphi_{1*}$ be represented by matrices $\alpha=({\alpha_{ij}})_{l\times p}$ and $\beta=({\beta_{ij}})_{l\times p}$, and let $\varphi_{0*}',\varphi_{1*}'$ be represented by matrices $\alpha'=({\alpha_{j'}'})_{1\times p'}$ and $\beta=({\beta'_{j'}})_{1\times p'}$.

Since $Sp(B)=Sp(F_1')\cup (0,1)$, we can choose a base point $x_0\in (0,1)$, Set $\phi_{x_0}=\pi_{x_0}\circ \phi$, denote $Sp\phi_{x_0}$ by $$Sp\phi_{x_0}=\{\theta_1^{\sim t_1},\theta_2^{\sim t_2},\cdots,\theta_p^{\sim t_p},y_1,y_2,\cdots,y_{\bullet}\}$$ where $y_1,y_2,\cdots,y_{\bullet}\in \coprod_{i=1}^{l}(0,1)_i\subset Sp(A)$.

$Step\,1$: We will use the technique in lemma 2.21 of [@EG] to construct disjoint closed intervals $V_0^i,V_1^i,V_2^i,\cdots,V_K^i$ in $[0,1]_i\subset Sp(A)$ for $i=1,2,\cdots,l$ with the following properties:

\(i) Define closed sets $W_r^i$ by $$W_r^i=\{x\in [0,1]_i,\quad d(x,V_r^i)\leq2\eta=\frac{1}{4K(L+1)}\}.$$ One has $W_{r_1}^i\cap W_{r_2}^i=\varnothing$ if $r_1\neq r_2$.

\(ii) Diameter $(W_r^i)<\dfrac{2}{K}$ for each 