proceed as in case [**(A)**]{} as there is no such $\delta>0$, that $|x_i(0)-x_j(0)|>\delta$. However for this situation we have prepared Lemma \[equi\], which implies uniform H" older continuity of $v^n$ in some neighborhood of $0$. Therefore for sufficiently small $t_0$ and $j\in B_i(0)$, such that $x_j(0)=x_i(0)$, we have $$\begin{aligned}
|x^n_i(s)-x^n_j(s)|\geq s\left(|v_i(0)-v_j(0)|-2Ls^{1-\alpha}\right)\geq s\frac{1}{2}|v_i(0)-v_j(0)|=:\delta_s>0\end{aligned}$$ for $s\in [0,t_0]$. On the other hand, for all $j\in B_i(0)$ such that $x_j(0)\neq x_i(0)$, from property 4 of Proposition \[properties\], we have $$\begin{aligned}
|x^n_i(s)-x^n_j(s)|\geq |x_i(0)-x_j(0)|-2Ms\geq \delta_s\end{aligned}$$ for all $n=1,2,...$ and all $s\in[0,t_1]$ with $0<t_1<1$ possibly smaller than $t_0$. Thus in $t_1$ we end up in a situation from case [**(A)**]{} with $$\begin{aligned}
|x^n_i(t_1)-x^n_j(t_1)|\geq\delta_{t_1}\end{aligned}$$ and all velocities $v^n$ are uniformly H" older continuous on $[0,t_1]$ and uniformly Lipschitz continuous on $[t_1,t]$ for all $t<T_0$. Again by Arzela-Ascoli theorem, there exists a $(C^1([0,t]))^{Nd}$ convergent subsequence of $x^n$.

\[subseq\] As in Remark \[sususus\], even though $x$ from Proposition \[odc\] is a limit of some subsequence of $x^n$, we will assume that it is in fact a limit of the whole sequence $x^n$ (by restricting the approximate solutions to only those, which approximate $x$). Such assumption will pose no threat to our reasonings as long as they will not involve uniqueness 