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[en] As the products of the supernova explosion SN 1054, both the Crab pulsar and its surrounding nebula harbour powerful sites of particle acceleration. They accelerate relativistic electrons which can be observed via their leptonic synchrotron and inverse-Compton emission. Such a pulsar/nebula complex is firmly detected from radio to -ray bands. For the Crab Nebula, the transition from synchrotron-dominated to inverse-Compton-dominated emission occurs at 1 GeV. Through analyses on Fermi Large Area Telescope -ray data accumulated over at least 9 years of observations, we investigate the GeV-TeV spatial morphology of the inverse-Compton nebula, the temporal variability of the synchrotron nebula in the tens to hundreds of MeV energy range, and the MeV-GeV spectra of the Crab pulsar. Comparisons of our results with ground-based instruments' observations from super-GeV to TeV photon energies allow us to interpret the emission mechanisms more comprehensively. We found that the spatial extension of the nebular inverse-Compton emission shrinks with increasing photon energy ( E where = 0.1550.035-0.037). Such a strong energy-dependence deviates from the model prediction for the dominating Thomson scattering, under an assumption of a spatially uniform seed photon field and a homogeneous magnetic field. The especially large extensions in 5-20 GeV imply that the external inverse-Compton emission is non-negligible, in addition to the synchrotron-self-Compton emission. For the synchrotron component of the Crab Nebula, in addition to confirming the flaring behaviour, we discovered a -ray low-flux state with a transition time of at most ten days. This indicates that the bulk (at least three-fourth) of the synchrotron emission above 100 MeV originates in a compact volume with an apparent angular size of 0''.4 t/(5 d) for a given timescale of transitions between low-flux and intermediate states t. Specifically, the inner-knot feature observed near the pulsar position is discussed as a possible candidate. For the Crab pulsar's -ray emission, we found an energy-dependent pulse shape and a phase-dependent spectral shape, which probably imply a multi-origin scenario involving the polar-cap, outer-gap, and relativistic-wind regions. We propose that these three acceleration sites dominate the emissions at different phases and energies respectively. Noteworthily, we detected a relatively sharp cutoff at a relatively high energy of 8 GeV for the bridge-phase emission, and the > 10 GeV spectrum for the second pulse peak is observed to be harder than those for other phases.