Derivation and New Interpretation of the Lorentz Transformations and Einstein's Theorem of Velocity Addition

It is traditionally believed that the Lorentz transformations (LT) and Einstein's theorem of velocity addition (ETVA), underlying special relativity, cannot be obtained from non-relativistic (classical) mechanics. In the present paper it is shown, however, that both the LT and the ETVA are derivable within the framework of classical kinematics if the speeds of material points are bounded above by a certain universal limit $c_+$ which can coincide with the speed of light $c$ in a vacuum.