On the Original Proof by Reductio ad Absurdum of the Hohenberg-Kohn   Theorem for Many-Electron Coulomb Systems
  It is shown that, for isolated many-electron Coulomb systems with Coulombic external potentials, the usual reductio ad absurdum proof of the Hohenberg-Kohn theorem is unsatisfactory since the to-be-refuted assumption made about the one-electron densities and the assumption about the external potentials are not compatible with the Kato cusp condition. The theorem is, however, provable by more sophisticated means and it is shown here that the Kato cusp condition actually leads to a satisfactory proof. 
Eugene S. Kryachko