The second Hohenberg-Kohn theorem is a consequence of the first, since it is necessary that the external potential (which in this case is the observable that gives the energy) being a unique functional of the density of the ground state and that the functional which gives the minimum energy in the ground state is obtained from the input density of the true ground state density. the density mentioned in the first Theorem will be the true density necessary to obtain the minimum ground state energy of the second Theorem.