Another Superluminal Thought Experiment

Suppose we have two particles $A$ and $B$ that fly in the opposite direction from the fixed point $O$, with the speeds $v_{1}$ and respectively $v_{2}$ with respect to an observer that stays in the point $O.$ Let's consider that $v_{1} + v_{2} \ge c.$ \begin{itemize} \item But, an observer that travels with particle $A$ (therefore he is at rest with particle $A)$ measures the speed of particle $B$ as being $v = \quad v_{1} + v_{2} \ge c.$ \end{itemize} Similarly for an observer that travels with particle $B$: he measures the speed of particle $A$ as also being superluminal: $v = \quad v_{1} + v_{2} \ge c.$ \begin{itemize} \item If we suppose $v_{1} = c$ and $v_{2}$\textit{ \textgreater 0}, then for the observer that travels with particle $A$ his speed with respect to observer in $O$ is $c$. But, in the same time, for the observer that travels with particle $A$ his speed with respect to particle $B$ should be greater that $c$, otherwise it would result that particle $B$ was stationary with respect to observer in $O$. It results that $c + v_{2}$\textit{ \textgreater c }for non-null$ v_{2}$, contrarily to the Special Theory of Relativity. \end{itemize}