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A comprehensive eigenchannel R-matrix study of photodetachment of He-(1 s2s2p 4P0 (1 ) is presented for the energy region of the Hesnl3l) thresholds, where 3<~n<~5. Over this energy region we present all partial cross sections which have been measured experimentally, including theoretical predictions (1 for the Hes3p 3P ( ) partial cross section, as well as energies and widths for all resonances, and identifications of all major ones. Near the Hen=3) thresholds, we predict two 4De resonances that to our knowledge have previously been neither predicted nor observed, and present results showing their effects on partial cross sections and photoelectron angular distributions; our He (1s2s 3S) and He (1s2p 3P) partial cross-section results are shown to be in excellent agreement with relative measurements of Klinkmüller and co-workers [Phys. Rev. A56 , 2788 (1997); J. Phys. B31, 2549 (1998)]; our predicted partial cross sections and resonance energies and widths are also compared with other theoretical predictions. Near the He (n=4 and 5) thresholds, we predict about 30 quartet Feshbach resonances and four quartet shape resonances which have not been observed or predicted (1 before; our predicted Hes3p 3P) partial cross section is in excellent agreement with measurements of Kiyan et al. [Phys. Rev. Lett. 81, 2874 (1998)], and we present identifications for all but one of the experimentally observed resonance features; our resonance energies and widths are also compared with other theoretically predicted results. Because resonances in the energy region considered have in general only small effects on the total photodetachment cross section, we show that alternative partial cross sections for He (1snl3l ) states tend to mirror each other in the vicinity of a resonance, as predicted analytically recently by us [Phys. Rev. A59, R1731 (1999)]. An exceptional case is found, however, for several 4Pe window resonances in the vicinity of the He (n=4) thresholds: the partial cross sections in the vicinity of these three resonances mimic each other. We prove analytically in an appendix that such mimicking behavior occurs whenever the Fano-Cooper profile parameter ρ2 [Phys. Rev. A137, A1364 (1965)] tends to unity.