Trigonometric Form of Complex Numbers

One can prove the following propositions: (1) Let F be an add-associative right zeroed right complementable left distributive non empty double loop structure and x be an element of the carrier of F . Then 0F · x = 0F . (2) Let F be an add-associative right zeroed right complementable right distributive non empty double loop structure and x be an element of the carrier of F . Then x · 0F = 0F . The scheme Regr without 0 concerns a unary predicate P, and states that: P[1] provided the parameters meet the following conditions: • There exists a non empty natural number k such that P[k], and • For every non empty natural number k such that k 6= 1 and P[k] there exists a non empty natural number n such that n < k and P[n]. One can prove the following propositions: (3) 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 the carrier of CF holds R(z) ­ −|z|. (6) For every element z of the carrier of CF holds I(z) ­ −|z|.