Abstract: We consider finite orthonormal systems and pose for them the question about strengthening of the classical Lieb-Thirring inequality. We prove such Lieb-Thirring-type inequalities and demonstrate that they are useful in applications. In particular, we establish estimates for the eigenvalue sums of some self-adjoint Schrodinger operators bounded from below for which the discreteness of the spectrum is verified by the classical Molchanov's criterion.