The theory of symmetric polynomials of multiple variables produces an unlimited set of algebra identities. These formulas contain numerous coefficients. For the most part their evaluation is difficult. This article presents a certain method of finding the coefficients which is based on some theorems of the theory. As a result, we obtain a linear system with only one solution. This method has been checked for the degree not greater that 6. Hypothetically, we can assume that the method is correct for any natural degree
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