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

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