The microscopic model of suppressing thermal smearing of electrons in resonant tunnelling structures with nonlinearity due to electron-phonon interaction under the assumption of the presence of degenerate levels in the quantum well is proposed. The electron-phonon interaction results in an effective attraction with breaking the degeneracy and lowing the energy level. We note the manifestation in this model of the Fano effect associated with multichannel tunnelling. As an example, the effect of suppressing the thermal spread of electrons in a nanotransistor with a metal quantum dot is considered and compared with the experiment. A rather significant (about 200 K) heating of the quantum dot was detected caused by the passage of a current through the nanotransistor. Despite heating, the effect of cold electrons filtration is clearly observed. The theory and experiment were found to be in a good agreement.