Abstract
In an n-element set of$X^{n}$positive integers, find a$k$element subset power$X^{k}$, the sum of the elements of which is equal to the certificate$S$. Lemmas on the polynomial solvability of the sum of subsets problem are proposed. Lemma 1 defines the relationship between the cardinalities of a set and a subset. Their ranges of variation are found using the polynomial formulation of the sum of subsets problem and the Waring's hypothesis for polynomials. Lemma 2 shows the possibility of representing some certificate as a sum of elements from some subset based on the Classical Waring Problem in number theory. The proved lemmas and completeness of the subset sum problem provide solutions to other$NP$- complete class problems and the validity of the equality of classes$P=NP$(a well-known theorem: if some$NP$- complete problem is solvable in polynomial time, then$P=NP$.