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Trigonometric form of complex numbers
 Journal of Formalized Mathematics
"... The scheme Regr without 0 concerns a unary predicateP, and states that: P[1] provided the parameters meet the following requirements: • There exists a non empty natural number k such thatP[k], and • For every non empty natural number k such that k � = 1 andP[k] there exists a non empty natural numbe ..."
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The scheme Regr without 0 concerns a unary predicateP, and states that: P[1] provided the parameters meet the following requirements: • There exists a non empty natural number k such thatP[k], and • For every non empty natural number k such that k � = 1 andP[k] there exists a non empty natural number n such that n < k andP[n]. The following propositions are true: (3) 1 For every element z of C holds ℜ(z) ≥ −z. (4) For every element z of C holds ℑ(z) ≥ −z. (5) For every element z of CF holds ℜ(z) ≥ −z. (6) For every element z of CF holds ℑ(z) ≥ −z. (7) For every element z of CF holds z  2 = ℜ(z) 2 + ℑ(z) 2. (8) For all real numbers x1, x2, y1, y2 such that x1 + x2iCF = y1 + y2iCF holds x1 = y1 and x2 = y2. (9) For every element z of CF holds z = ℜ(z)+ℑ(z)iCF. (10) 0CF = 0+0iCF. (12) 2 For every unital non empty groupoid L and for every element x of L holds power L (x, 1) = x. (13) For every unital non empty groupoid L and for every element x of L holds power L (x, 2) = x · x. (14) Let L be an addassociative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n> 0, then power L (0L, n) = 0L. 1 The propositions (1) and (2) have been removed. 2 The proposition (11) has been removed.
Primitive roots of unity and cyclotomic polynomials
 Journal of Formalized Mathematics
"... Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials. ..."
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Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.
Hermitan functionals. Canonical construction of scalar product in quotient vector space
 Formalized Mathematics
"... Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The ..."
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Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The construction of scalar product in quotient vector space from non negative hermitan functions is the main result of the article.