In this article, based on a recent theorem by Lieb er al, we shall prove two theorems on Ihe momentum distribution functions of the half-filled Hubbard model on a ddimensional simple cubic lattice in a mathematically rigorous way. More precisely, we shall first show that the half-filled positive-U and negative4 Hubbard models have the same momentum distribution functions n (4) and n+(q). Then, we will show that n.(q) are symmetric functions about the value E = 3. Fmally. we shall briefly discuss some possible applications of these theorems to the further numerical investigations on the ground state of the Hubbard model at half-filling.