I have the following problem:

Example: collision of 2 electrons

For non-relativistic scattering it is easy to show that the speed of the CM frame with respect to the lab frame is equal to the speed of the electrons in the CM frame, expoloiting the fact that in the lab frame, one of the electrons is at rest.

Now this also holds in the relativistic regime. I'm not really sure where does this follow from. Is it valid to take the lorentz velocity addition formula, and taking the speed of the particle in the lab frame to be 0 we see that for x-coordinate the speed of the electron in CM frame must then be uqual to to the speed of CM frame in lab frame. Saying that the lab and CM frames are in standard configuration makes this proof general enough?