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Actually no, the 2nd HK theorem is not a consequence but its directly connected with the 1st HK theorem. The fact that there is a one-to-one correspondence of the Vext with the density, let the 2nd HK theorem ensure that if you actually apply the ground state functional to any other density that is not the ground state, necessarily it will have a higher energy (because of the bijective nature and because the original density is the lowest possible for that functional)
So, they are not a consequence but they are complemented.
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