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Strong electric discharges associated with thunderstorms can produce terrestrial gamma-ray flashes (TGFs), i.e., intense bursts of x rays and γ rays lasting a few milliseconds or less. We present in this Letter new TGF timing and spectral data based on the observations of the Italian Space Agency AGILE satellite. We determine that the TGF emission above 10 MeV has a significant power-law spectral component reaching energies up to 100 MeV. These results challenge TGF theoretical models based on runaway electron acceleration. The TGF discharge electric field accelerates particles over the large distances for which maximal voltages of hundreds of megavolts can be established. The combination of huge potentials and large electric fields in TGFs can efficiently accelerate particles in large numbers, and we reconsider here the photon spectrum and the neutron production by photonuclear reactions in the atmosphere.