Hohenberg-kohn 2nd

No it is not a contunition, while 1ste theorem proves a bijective relation between psi and density, is doesnt tell anything about the density or psi in relation to the external potential. While the second theorem tells us that the energy functional is minimized with the true ground state density. But I think in the prove u need to prove it with psi not the density and then u can use 1st theorem to extend it further to the density. So the overal 2nd theorem is fundamentally differant but, it extends to density by the continition using the 1st theorem