Theorem:
Let, $$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]\left(y\right)=0$$ be a fractional differential equation of order $\left(n,q\right)$, where $v=\frac{n}{q}$, and let $$P\left(x\right)=x^{n}+a_{1}x^{n-1}+\dots+a_{n}$$ be the corresponding indicial polynomial.
Let $$y_{1}\left(t\right)=\mathscr{L}^{-1}\left\{ P^{-1}\left(s^{v}\right)\right\} $$ Then if $N$ is the smallest integer with the property $N\geq n\cdot v$, then $$y_{1}\left(t\right), y_{2}\left(t\right)=D^{1}y_{1}\left(t\right),...,y_{N}\left(t\right)=D^{N-1}y_{1}\left(t\right)$$ are $N$ independent solutions.
This theorem is from the book "An Introduction to Fractional Calculus and Fractional Differential Equations" by Miller and Ross. I don't manage to understand the proof provided in the book.
Proof:
We take the Laplace transform of the equation:
$\mathscr{L}\left\{ P\left(D^{v}y\left(t\right)\right)\right\} =0$,
and also $$\mathscr{L}\left\{ P\left(D^{v}y\left(t\right)\right)\right\} =P\left(s^{v}\right)Y\left(s\right)-\sum\limits_{r=0}^{N-1}B_{r}\left(y\right)s^{r}$$ where $B_{r}\left(y\right)$ is a linear combination of terms of the form:
$$D^{kv-\left(r+1\right)}y\left(0\right)$$
$k=rq+1,\dots,n$ $r=0,1,\dots,N-1$
In particular $B_{0}\left(y\right)=P\left(D^{v}\right)D^{-1}y\left(0\right)-a_{n}D^{-1}y\left(0\right)$.
We have,
$$Y\left(s\right)=\frac{\sum\limits_{r=0}^{N-1}B_{r}\left(y\right)s^{r}}{P\left(s^{v}\right)}$$ and $$y\left(t\right)=\mathscr{L}^{-1}\left\{ Y\left(s\right)\right\} $$ is a solution of the equation.
Now let,
$y_{1}\left(t\right)=\mathscr{L}^{-1}\left\{ P^{-1}\left(s^{v}\right)\right\} $, then
$\mathscr{L}\left\{ P\left(D^{v}\right)y_{1}\left(t\right)\right\} =P\left(s^{v}\right)-\sum\limits_{r=0}^{N-1}B_{r}\left(y_{1}\right)s^{r}$.
The initial value theorem for the Laplace transform says that, $\underset{s\rightarrow+\infty}{\lim}s^{v+1}\mathscr{L}\left\{ f\left(t\right)\right\} =D^{v}f\left(0\right)$ for all $v\in\mathbb{R}$. Thus, $B_{0}\left(y_{1}\right)=1$, $B_{r}\left(y_{1}\right)=0$ for $r>1$
Is this last deduction in italics where I'm stuck. It is not clear to me.