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Comprehensive studies of the low-temperature specific heat C(T,H) in NaxCoO2-yH2O are presented. At H=0, a very sharp anomaly was observed at T=4.7 K indicating the existence of bulk superconductivity. There exists the T^2 term in C(T,H=0) in the superconducting state manifesting the line nodal superconducting order parameter. The superconducting volume fraction is estimated to be 26.6 % based on the consideration of entropy conservation at Tc for the second-order superconducting phase transition. An abrupt change of the slope in Tc(H) was observed. Possible scenarios such as the multiple phase transitions in the mixed state are discussed.

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