Is theorem 2 a consequence of theorem 1?

I think it depends on what you would understand under “a natural consequence”, as I think you could technically describe the proof of any theorem as a natural consequence of other theorems. The first KH theorem tells us that any observable can be written as a functional of the electron density, but the second terms adds to this that the ground state energy is lower than the expectation value of the energy for any other trial wave function, and thus the energy functional must be minimal for the ground state density.