we have $|v^n_k(0)-v^n_i(0)|> 0$, thus $$\begin{aligned}
    |v^n_k(s)-v^n_i(s)|\geq |v^n_k(0)-v^n_i(0)|-2L|s|^{1-\alpha},\end{aligned}$$ which is positive for $s\leq t_0$ with $t_0$ defined by (\[tn1\]).

3.  To prove that $\xi$ has a constant sign in $[0,t_n]$ we applied Lemma \[equitech\] concluding that $|\xi(s)|$ is positive, provided that $s$ belongs to the interval on which the thesis of Lemma \[equitech\] holds and we proved above that this interval includes $(0,t_0]$.

Therefore all bounds from points 1,2 and 3 are satisfied for $t_0$ defined by (\[tn1\]) and it is clearly $n$-independent. Thus we proved that there exists an interval $[0,t]$ with $t\geq t_0$ in which all functions $v_i^n$ are uniformly H" older continuous.

Definition of the solution
--------------------------

Before we define the solution let us recall property 6 from Proposition \[properties\], which basically states that the trajectories of the particles cannot separate if they stick together at some point. This is an obvious consequence of the uniqueness for the approximate solutions. However, since $\psi$ is singular at $0$ it may happen that the solutions of the (C-S) model with $\psi$ are not unique and that the trajectories may split as in the case of the well known example $\dot{y}=cx^\frac{1}{3}$. In fact a loss of uniqueness may happen at 