The 1st HK theorem tells that ther’s a one-to-one correspondence between the ground state density of a many-electron system and the extrernal potential. When looking at the Schrödinger’s equation you see that there’s a relation between potential and energy. However, it’s not clear what the correspondence between V and rho is (I think?). So I think telling the 2nd theorem is a natural consequence of the 1st isn’t right.