reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th51:
  for a being Complex holds |.Rotate(a,r).| = |.a.|
proof
  let a be Complex;
  Re Rotate(a,r) = |.a.|*cos (r+Arg a) & Im Rotate(a,r) = |.a.|*sin (r+
  Arg a) by COMPLEX1:12;
  hence |.Rotate(a,r).| = sqrt((|.a.|*cos (r+Arg a))^2 + (|.a.|*sin (r+Arg a))
  ^2) by COMPLEX1:def 12
    .= sqrt(|.a.|^2*((cos (r+Arg a))^2 + (sin (r+Arg a))^2))
    .= sqrt(|.a.|^2*1) by SIN_COS:29
    .= |.a.| by COMPLEX1:46,SQUARE_1:22;
end;
