Polynomial Solvability of the Subset Sum Problem

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
$P=NP$
.