We have investigated the process of protein folding by Monte-Carlo simulation of folding occurring in a simple 3D lattice model of a protein globule. We have found the range of 'optimal' temperatures where the native fold is achieved by the Monte-Carlo process much faster than that by exhaustive sorting of all the chain folds. The 'optimal' temperatures are essentially the same for different random and 'edited' sequences (for the latter, the native fold energy is separated by a considerable gap from the energies of other low-energy folds; for random sequences, this gap is negligible). At the 'optimal' temperatures, the 'edited' chains attain their native fold faster than the random ones. However, the essence is that the native folds of 'edited' chains are thermodynamically stable at temperatures optimal for fast folding, while the native folds of random chains are unstable at the temperatures optimal for fast folding; also, at low temperatures where the native folds of random chains are stable, folding kinetics is very slow. Consequently, stable native folds are formed slowly by random sequences and rapidly by the 'edited' ones.