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The first theorem tells us there is a unique density that exists for an external potential and/or a wave function. It doesn’t provide us any information about the identity, or how to find the particular density if it exists. The second theorem shows us a direction to find out the unique density. While is the idea that it might be the lowest density is a natural consequence of human thought, it need not be the case according to the rules of nature. It is a separate independent theorem.
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