every projection in $\mathfrak{A} / \mathfrak{I}$ lifts to a projection in $\mathfrak{A}$, then there exists a $*$-homomorphism ${  \phi  \colon  \mathfrak{A} / \mathfrak{I}  \rightarrow  \mathfrak{A}  }$ such that $\pi \circ \phi = {\ensuremath{\operatorname{id}}}_{ \mathfrak{A} / \mathfrak{I} }$, where ${  \pi  \colon  \mathfrak{A}  \rightarrow  \mathfrak{A} / \mathfrak{I} }$ is the quotient map.

Consequently, if $\mathfrak{A}$ is a $C^{*}$-algebra with an ideal $\mathfrak{I}$ such that for every projection $p\in \mathfrak A$ the corner $p\mathfrak{I}p$ has an approximate identity consisting of projections, $\mathfrak{A} / \mathfrak{I}$ is an AF-algebra, and every projection in $\mathfrak{A}/\mathfrak{I}$ lifts to a projection in $\mathfrak{A}$, then there exists a sequence of finite dimensional sub-$C^{*}$-algebras $\{ \mathfrak{C}_{k} \}_{k = 1}^{\infty}$ of $\mathfrak{A}$ such that $\mathfrak{C}_{k} \cap \mathfrak{I} = 0$ for all $k$, $\mathfrak{C}_{k} + \mathfrak{I} \subseteq \mathfrak{C}_{k+1} + \mathfrak{I}$ for all $k$, and $\bigcup_{ k = 1}^{\infty} ( \mathfrak{C}_{k} +\mathfrak{I} )$ is dense in $\mathfrak{A}$.

The first part of the lemma is proved in the same way as [@dim Lemma 9.8]. The following are the key ingredients of the proof: (1) the existence of an approximate identity consisting of projections for $p \mathfrak{I} p$ for every projection $p \in \mathfrak{A}$ and (2) every projection in $\mathfrak{A}/ \mathfrak{I}$ lifts to a 