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The explanation on wikipedia is more mathematically focused, highlighting that if two systems of electrons have the same density, then their external potential must be differ by a constant.
This is a bit less intuitive compared to the the explanation in the video, where we go from subjective to injective relations. this focuses more on the fact that the density of the system is also injective.
In both cases, we conclude that the density contains all information of the external potential of the system.
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