The analysis discussed in this study and its previous version is based on a well-known NP-complete problem which is called the "satisfiability problem" or "SAT". From SAT a new NP-complete problem, called "core function", derives; this problem is described by a Boolean function of the number of the clauses of SAT. In this study, a new proof is presented according to which the number of gates of the minimal implementation of core function increases with n exponentially. Since the synthesis of the core function is an NP-complete problem, this result can be considered as the proof of the theorem which states that the class P of all the decision problems which can be solved in polynomial time does not coincide with the class NP of the problems for which an answer can be verified in polynomial time.