Let \(y\) be the solution of the Thomas-Fermi differential equation \(y'' =x^{1/2} y^{3/2}\); \(y(0)=1\); \(y(\infty)=0\). Let \(\Omega^ 2_ c=\sup_{r>0}u(x)\); where \(u(x)= xy(x)\). Let \[ F(\Omega)=\int \left( {y \over x}-{\Omega^ 2 \over x^ 2} \right)_ +^{1/2}dx, \quad \Omega \in (0,\Omega_ c). \] The main result of the work is a proof of the inequality \(F''(\Omega) \leq c<0\;\forall \Omega \in (0,\Omega_ c)\). Computer aided analysis and a variety of conventional lemmas are supposed to prove this result in 143 pages. This result together with those in a further half-a-dozen papers to be published shortly will provide a proof of the asymptotic formula for the ground state energy of a nonrelativistic atom announced a few years ago by the same authors [Bull. Am. Math. Soc. 23, 525-530 (1990; Zbl 0722.35072)].