Consistency of Equations in the Second-order Gauge-invariant Cosmological Perturbation Theory

Along the general framework of the gauge-invariant perturbation theory developed in the papers [K. Nakamura, Prog. Theor. Phys. {\bf 110} (2003), 723; {\it ibid}, {\bf 113} (2005), 481.], we re-derive the second-order Einstein equations on four-dimensional homogeneous isotropic background universe in gauge-invariant manner without ignoring any mode of perturbations. We consider the perturbations both in the universe dominated by the single perfect fluid and in that dominated by the single scalar field. We also confirmed the consistency of all equations of the second-order Einstein equation and the equations of motion for matter fields which are derived in the paper [K. Nakamura, arXiv:0804.3840 [gr-qc]]. This confirmation implies that the all derived equations of the second order are self-consistent and these equations are correct in this sense.