which is a contradiction identical to the one found in Hohenberg and Kohn's original proof. From Lemma 1, we establish that for every eigenspace of any Hamiltonian operator there is a unique a eigendensity associated with it. This eigendensity can be easily computed when an orthonormal basis of the corresponding eigenspace is known. Consequently, the existence of the map f := Ĥ → ρ ε 0 is affirmed. Additionally, the reciprocal mappings of any Hamiltonian operator to its external potential and vice versa are facilitated by the functions: