I wouldn’t say it is a natural consequence, instead it builds on the first one and helps us further to solve the problem at hand. The first theorem states that external potentials, their corresponding wavefunction solutions and the ground state densities corresponding to this set of wavefunctions are in bijection. The second theorem tells us that there exists a functional on densities, which gives us the total energy of the system with the corresponding external potential if we plug in the ground state density, and even further the the functional reaches its minimum in the ground state density. Thus the problem of finding the total energy for a given ground state density reduces to minimizing this functional (if we can find an expression for this functional). Thus, the second theorem gives us, given this functional, a way to find the total energy given the ground state density, or to find the ground state density by minimizing this functional. Thus in essence, given this functional, we have means to solve for the ground state density, which has all the information of the system.