  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.
