Let $A$ be a $C^*$-algebra, we denote with $V(A)$ the semigroup of Murray-von Neumann equivalence classes of projections in matrices over $A$ (as usual). In https://arxiv.org/pdf/math/0310340.pdf, above definition 5.1. in section 5 (on page 20), there is mentioned the following fact:

If $A$ is a $C^*$-algebra and $p$ a projection in $A$, and $n\in\mathbb{N}_{\ge 1}$, then $nx=[p]$ has a solution $x\in V(A)$ if and only if there is a unital embedding $M_n\to pAp$. $(\star)$

I need help to prove the direction $\Rightarrow$. (The reason why I take a closer look to this statement is that I need $\star$ for an other proof.)

To prove that there is such a unital embedding, one strategy could be (if I'm not mistaken): Find mutually orthogonal projections $f_1,\cdots f_n$ in pAp, with $f_1\sim f_2 \sim \cdots \sim f_n$ (where $\sim$ is Murray-von Neumann equivalence) and such that $1=f_1+ \cdots +f_n$. Then you can extend the set $\{f_1,\cdots,f_n\}$ to matrix units of $M_n$.

Now, if $nx=[p]$ for an $x\in V(A)$. Write $x=[x']$ for a projection $x'\in M_m(A)$ for some natural number $m$. Then $\operatorname{diag}(x',x',\cdots,x')\sim_0 p.$ I.e. if $p$ is a projection in $M_k(A)$, then there exists a partial isometry $v\in M_{k,nm}$ such that $\operatorname{diag}(x',x',\cdots,x')=vv^*$ and $p=v^*v$ (it's Murray-von Neumann equivalence in different sized matrices). But I don't know how to get these projections $f_i$ (from here)...

How to get these projections $f_i$/ or can you provide an alternative approach to prove it/ or can you provide a reference for a proof of $\star$? Thank you.