
theorem Th23:
  for n being non zero Element of NAT, x being Element of
  F_Complex st x in n-roots_of_1 holds |.x.| = 1
proof
  let n be non zero Element of NAT, x be Element of F_Complex such that
A1: x in n-roots_of_1;
A2: now
    assume x = 0.F_Complex;
    then (power F_Complex).(x,n) <> 1_F_Complex by VECTSP_1:36;
    then not x is CRoot of n, 1_F_Complex by COMPLFLD:def 2;
    hence contradiction by A1,Th21;
  end;
  then
A3: |.x.| > 0 by COMPLFLD:59;
  x is CRoot of n,(1_F_Complex) by A1,Th21;
  then power(x,n) = 1_F_Complex by COMPLFLD:def 2;
  then
A4: 1 = |.x.| to_power n by A2,COMPLFLD:60,POLYNOM5:7;
  assume
A5: |.x.| <> 1;
  per cases by A5,XXREAL_0:1;
  suppose
A6: |.x.| < 1;
    reconsider n9 = n as Rational;
    |.x.| #Q n9 < 1 by A3,A6,PREPOWER:65;
    hence contradiction by A4,A3,PREPOWER:49;
  end;
  suppose
    |.x.| > 1;
    hence contradiction by A4,POWER:35;
  end;
end;
