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 $e\in A$,

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

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

we consider $\pi^{i'}\circ \phi$ instead of $\phi$ and repeat the four steps in \[dc\]. Note that in this case $Sp(B)=Sp(F_1')\cup$ $\coprod_{i'=1}^{l'}(0,1)_{i'}$, then for each $i'\in\{1,2,\cdots,l'\}$, we will choose $(x_0,i')\in (0,1)_{i'}$ be the base points, then we can prove the result in the same way, so we omit this proof.

Let $A=\underrightarrow{lim}(A_n,\phi_{m,n})$ be a ${\mathrm{C}}^*$-algebra which is an inductive limit of the Elliott-Thomsen building blocks $A_m$ with morphisms $\phi_{m,n}:A_m\rightarrow A_n$. If RR(A)=0, then for any finite subset $G\subset A_m$, $\varepsilon>0$, $L>0$, there exists $n\geq m$, such that there are two orthogonal projections $q_1,q_2\in A_{m}$ and a unital homomorphism $\phi':A_m\rightarrow q_2A_nq_2$ with finite dimensional image, satisfying the following:

$(1)\,\,{\rm rank}\,\,q_2\geq L\cdot{\rm rank}\,\,q_1,\,\,where\,\,q_1+q_2=\phi_{m,n}(1_{A_m})$,

$(2)\,\,\|q_1\phi(g)-\phi(g)q_1\|<\epsilon\,\,\,for\,\,any\,\, g\in G,$

$(3)\,\,\|\phi_{m,n}(g)-q_1\phi_{m,n}(g)q_1\oplus\phi'(g)\|<\epsilon \,\,\,for\,\,any\,\,g\in G.$

[**Acknowledgement**]{}

The author would like to thank Guihua Gong for helpful suggestions and Yuanhang Zhang for helpful discussions.

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