  For a dilute system of non-relativistic bosons interacting through a positive
potential $v$ with scattering length $a$ we prove that the ground state energy
density satisfies the bound $e(\rho) \geq 4\pi a \rho^2 (1+
\frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} +o(\sqrt{\rho a^3}\,))$, thereby
proving a lower bound consistent with the Lee-Huang-Yang formula for the energy
density. The proof allows for potentials with large $L^1$-norm, in particular,
the case of hard core interactions is included. Thereby, we solve a problem in
mathematical physics that had been a major challenge since the 1960's.
