A new computationally efficient learning algorithm for a hybrid system called further Neuro-Fuzzy Kolmogorov's Network (NFKN) is proposed. The NFKN is based on and is the development of the previously proposed neural and fuzzy systems using the famous superposition theorem by A.N. Kolmogorov (KST). The network consists of two layers of neo-fuzzy neurons (NFNs) and is linear in both the hidden and output layer parameters, so it can be trained with very fast and simple procedures. The validity of theoretical results and the advantages of the NFKN in comparison with other techniques are confirmed by experiments.