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The first theorem states that there is a bijection between the electron density and the total energy. Therefore the ground-state density uniquely determines a potential, but only from the first theorem one cannot say that this is in fact the ground state energy. The second theorem does explicitly state that the ground-state density uniquely determines the ground state energy. However the first theorem is used in the proof of the second one (https://journals.aps.org/pr/pdf/10.1103/PhysRev.136.B864) so there is a link between the two.
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