All the 'proofs' of the Work–Energy Theorem that I have seen show that the work done by the resultant force acting on a body is equal to $\Delta \left(\tfrac 12 m v^2)\right)$ for that body. [It's easy to modify the proof to show that, for speeds not negligible compared with $c$, $\tfrac 12 m v^2$ needs to be replaced by $mc^2 (\gamma (v)-1)$.]
Such a derivation can correctly be called 'a proof'. But the usual practice is to call it a proof that the work done by the resultant force is equal to the increase in kinetic energy. This seems questionable; for what right do we have to call $\tfrac 12 m v^2$ or $mc^2 (\gamma (v)-1)$, 'kinetic energy'?
This particular objection disappears if kinetic energy is defined as $\tfrac 12 m v^2$ or $mc^2 (\gamma (v)-1)$, independently of any connection with work (to avoid circularity). But surely we don't make such an unsupported definition.
Arguably a better alternative is to define a body's kinetic energy as the amount of work it does coming to rest, or as the amount of work needed to bring it to its present speed from rest. In which case the role of the Work–Energy theorem is to establish the formula for kinetic energy, rather than to show that work done = change in kinetic energy.
I'd be interested in others' views as to the correct interpretation.
