an interval $[0,T_0)$ such that for any $[0,t]\in[0,T_0)$ solutions $x^n$ have a subsequence that converges in $(C^{1}([0,t]))^{Nd}$.

If for all $i,j=1,...,N$ initial velocity $v_i(0)=v_j(0)$ then by property 7 from Proposition \[properties\], we have $v^n\equiv v(0)$ for all $n$ and the assertion holds with $T_0=T$. From this point we assume that there exist $i,j=1,...,N$ such that $v_i(0)\neq v_j(0)$. Recall $B_i(0)$ defined by (\[bi\]) – the set of all indexes which are directly involved in the evolution of $v_i^n$ – from this point we will only consider $j\in B_i(0)$. There are two possibilities:

(A)

:   For all $i$ and $j\in B_i(0)$ we have $x_j(0)\neq x_i(0)$.

(B)

:   There exists $i$ and $j\in B_i(0)$ such that $x_j(0)=x_i(0)$.

[**(A)**]{} In this case there exists $\delta$ such that for all $i=1,..,N$ and all $j\in B_i(0)$ we have $|x_i(0)-x_j(0)|>\delta$ and by Remark \[t0\] for all $t<T_0$ there exist $\delta_t\in(0,\delta]$, such that for all $i,j$ we have $|x_i-x_j|>\delta_t$ on $[0,t]$, which implies that $\psi(|x_i-x_j|)\leq \delta^{-\alpha}$ and all velocities $v^n$ are uniformly Lipschitz continuous on $[0,t]$. Therefore by Arzela-Ascoli theorem there exists a $(C^1([0,t]))^{Nd}$ convergent subsequence of $x^n$.\
[**(B)**]{} In the second case, there exist $i$ and $j$, such that $x_i(0)=x_j(0)$ and $v_i(0)\neq v_j(0)$ and we may not 