of [@DNNP].

\[3.1\] Let $X$ be a Hausdorff compact space ,let $k'> k\geq 1$ be integers ,let $\mathcal{W}$ be an open cover of $X$, and assume that for each $W\in \mathcal{W}$ there is given a continuous projection valued map $p_W:W\rightarrow M_n$ such that ${\rm rank}\ p_W\geq k'$ for $x\in W$. If dim$(X)\leq2(k'-k)-1$, then there is a continuous projection valued map $p:X\rightarrow M_n$ such that for $x\in X$: $${\rm rank}\ p(x)\geq k,$$ $$p(x)\leq\bigvee\{p_W(x):W\in \mathcal{W},x\in W\}.$$

Let $A=A(F_1,F_2,\varphi_0,\varphi_1)\in\mathcal{C}$ be a minimal block, $p\in A$ be a projection, we will take $p$ as a projection in $C([0,1],M_{L(A)}(\mathbb{C}))$, where $L(A)=\sum_{i=1}^{l}l_i$, then rank $p\in \mathbb{N}$.

For a projection $p\in A=\oplus_{k=1}^{n}A_k$, where $A_1,A_2,\cdots,A_n\in \mathcal{C}$ are minimal blocks, let $\sigma_k:A\rightarrow A_k$ be a homomorphism which maps the direct sum to the $k^{th}$ block in a natural way, define $${\rm rank}\,\,p=({\rm rank}\,\,\sigma_1(p),{\rm rank}\,\,\sigma_2(p),\cdots,{\rm rank}\,\,\sigma_n(p))\in \mathbb{N}^n,$$ where we take $\sigma_k(p)$ as a projection in $C([0,1],M_{L(A_k)}(\mathbb{C}))$. Let $p,q\in A$ be two projections, we say that ${\rm rank}\,\,p\geq{\rm rank}\,\,q$, if ${\rm rank}\,\,\sigma_k(p)\geq{\rm rank}\,\,\sigma_k(q)$ holds for all $i\in \{1,2,\cdots,l\}$.

\[dc\] Let $A=A(F_1,F_2,\varphi_0,\varphi_1),B=B(F_1',F_2',\varphi_0',\varphi_1')\in \mathcal{C}$, assume that $F_2'$ has only one block. Let $G\subset A$ be a finite set, for any positive integers $J,L$, there exists $\eta>0$ such that if a unital homomorphism $\phi: 