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In a previous analysis of electroproduction of hypernuclei the cross sections were calculated in distorted-wave impulse approximation where the momentum of the initial proton in the nucleus was set to zero (the frozen-proton approximation). In this paper we go beyond this approximation assuming a non zero effective proton momentum due to proton Fermi motion inside of the target nucleus discussing also other kinematical effects. To this end we have derived a more general form of the two-component elementary electroproduction amplitude (Chew-Goldberger-Low-Nambu like) which allows its use in a general reference frame moving with respect to the nucleus-rest frame. The effects of Fermi motion were found to depend on kinematics and elementary amplitudes. The largest effects were observed in the contributions from the longitudinal and interference parts of the cross sections. The extension of the calculations beyond the frozen-proton approximation improved the agreement of predicted theoretical cross sections with experimental data and once we assumed the optimum on-shell approximation, we were able to remove an inconsistency which was previously present in the calculations.