= Solution
For fixed electron-electron interaction and electron number, a nondegenerate ground-state <electron density> determines the external potential up to an additive constant, and hence determines the Hamiltonian and ground-state observables.
Suppose two potentials $v,v'$ differing by more than a constant have different <ground states> $\Psi,\Psi'$ but the same density $\rho$. Strict <Rayleigh-Ritz variational principle> inequalities give
$$
E< E'+\int[v(\mathbf r)-v'(\mathbf r)]\rho(\mathbf r)\,d^3r,
$$
$$
E'< E+\int[v'(\mathbf r)-v(\mathbf r)]\rho(\mathbf r)\,d^3r.
$$
Adding produces the contradiction $E+E'<E+E'$. This proves the first <Hohenberg-Kohn theorem> under the stated nondegeneracy assumptions. A constant potential shift changes energy by $N$ times that constant without changing the density.<Inhomogeneous electron gas by Pierre Hohenberg and Walter Kohn>[Density-potential uniqueness argument]{ref}
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