In a way, the second theorem is a more specific statement of the first theorem. As I think we’ll later also handle in this course, it states that the external potential, which is the only input to the Kohn-Sham equations, can be uniquely determined from the electron density.

I believe, however, that the second HK theorem is a lot more powerfull than the first HK theorem. In physics, it can certainly happen that someone else improves your theory and adds their name to it, making it a new theory. Much like someone improving on an old theory, it’s only after realising the practicality of the second HK theorem, that one can realise it deserves to be a theorem on it’s own.