of $x$.

\[coroodc\] Let $x$ be as in Remark \[subseq\]. Then $x$ is a local classical solution to C-S system in the interval $(0,T_0)$. Moreover

1.  For all $i,j=1,...,N$, we have $|x_j-x_i|>0$ in $(0,T_0)$.

2.  The function $x$ is smooth in $(0,T_0)$.

By the definition of $T_0$ we get assertion 1, which on the other hand implies that in a neighborhood of each $t\in(0,T_0)$ all the derivatives of $x^n$ are uniformly bounded, which by Arzela–Ascoli theorem implies that $x$ is smooth in $(0,T_0)$. With this, to prove that $x$ solves C-S system with weight $\psi$, it suffices to take a $(C^2([t-\epsilon,t+\epsilon]))^{Nd}$ limit of systems associated with weights $\psi_n$, with $[t-\epsilon,t+\epsilon]\subset(0,T_0)$.

Our next step is to show that the function $x$ actually satisfies our problem in a weak sense in every interval $[0,t]\subset[0,T_0)$ (though to prove that it satisfies Definition \[sol\] we still nead continuity of $v$ at $T_0$).

\[spelnia\] For all $t\in[0,T_0]$ the function $x$ is a weak $(W^{2,1}([0,t]))^{Nd}$ solution of (\[cs\]).

From Proposition \[odc\] and Corollary \[coroodc\] we know that $x\in (C^1([0,T_0)))^{Nd}$ and that $T_0$ is the time of the first collision of the particles. It suffices to show that $x$ satisfies (\[cs\]) weakly in intervals $[0,t]$ for $t\nearrow 