How to derive the kinetic gas equation?

loom91

When he writes equation (1), he (correctly) assumes that the particles are moving with different velocities, but in the rest of the derivation he implicitly assumes that all particles are moving with same velocity, otherwise he can't write [tex]\delta p = N2mv_x[/tex]. Further, he uses the same label to identify this velocity and the variable velocity in the sum, effectively rendering the expressions logically meaningless. I never imagined that a derivation of such a common equation would be so difficult to find.