2nd HK theorem consequence of 1st HK theorem

We can deduct that the 2nd HK theorem is a direct consequence of the 1st HK theorem, because to demonstrate the 1st HK theorem we make use of the variational principle of quantum physics: we know that the Egs is the minimum energy of the system. In the 2nd theorem we use the bijective property between Psi and p: starting from po, we must conclude that the Egs must be the lowest energy.