some $M$ independent of $n$.

Point 5 follows immediately from property 3 and equation $(\ref{n})_2$, while points 6 and 7 are obvious consequences of uniqueness of the solutions.

\[setb\] Property 6 from the above proposition implies that the acceleration equation $(\ref{n})_2$ can be replaced by: $$\begin{aligned}
\label{cs3}
\dot{v}^n_i=\frac{1}{N}\sum_{k\in B_i(t)}(v^n_k-v^n_i)\psi_n(|x_k^n-x^n_i|),\end{aligned}$$ where $B_i(t)$ is defined by (\[bi\]), with $$\begin{aligned}
\dot{v}^n_i=0\end{aligned}$$ should set $B_i(t)$ be empty. This technical observation will be useful later on.

Hereinafter we will use $M(n)$ and $M$ in the same roles as in Proposition \[properties\]. We end this section with an important lemma that is in fact our way to deal with existence in a right sided neighborhood of a point of collision.

\[equi\] Let $x^n$ be a solution of C-S system on the time interval $[0,T]$ with weight $\psi_n$ and initial data $x(0),v(0)$ – independent of $n$. Then there exists an interval $[0,t]$, such that all velocities $v^n$ are uniformly H" older continuous on $[0,t]$.

To prove this lemma we need yet another, technical lemma.

\[equitech\] If $x_i(0)=x_j(0)$, then for all $n$, there exists an interval $(0,t_n]$, such that $$\begin{aligned}
|v_i^n(s)-v_j^n(s)|\leq 4\frac{|(v_i^n(s)-v^n_j(s))\cdot(x^n_i(s)-x^n_j(s))|}{|x^n_i(s)-x^n_j(s)|}\end{aligned}$$ for $s\in(0,t_n]$.

By property 5 from Proposition \[properties\], we have $$\begin{aligned}
\label{rn}
v^n_i(s)-v^n_j(s)=v^n_i(0)-v^n_j(0)+r_n(s),\end{aligned}$$ where $|r_n(s)|\leq 2|s|Mn$. 