
theorem Th6:
  for q being Real st q > 0 for r being Element of F_Complex st |.r
  .| = 1 & r <> [**1,0**] holds |.[**q, 0**] - r.| > q - 1
proof
  let q be Real such that
A1: q > 0;
  let r be Element of F_Complex such that
A2: |.r.| = 1 and
A3: r <> [**1,0**];
  set b = Im r;
  set a = Re r;
A4: a^2 + b^2 = 1^2 by A2,COMPTRIG:3;
A5: now
    assume
A6: a = 1;
    then b = 0 by A4;
    hence contradiction by A3,A6,COMPLEX1:13;
  end;
  a <= 1 by A2,COMPLEX1:54;
  then a < 1 by A5,XXREAL_0:1;
  then 2*q > 2*q*a by A1,XREAL_1:157;
  then -2*q*a > -2*q by XREAL_1:24;
  then -2*q + q^2 < -2*q*a + q^2 by XREAL_1:8;
  then
A7: q^2 - 2*q*a + 1 > q^2 - 2*q + 1 by XREAL_1:8;
  reconsider qc = [**q, 0**] as Element of F_Complex;
A8: Re[**q-a,-b**] = q-a & Im[**q-a,-b**] = -b by COMPLEX1:12;
  (|.qc - r.|)^2 = |.[**q, 0**] - [**a, b**].|^2 by COMPLEX1:13
    .= |.[**q - a, 0-b**].|^2 by POLYNOM5:6
    .= (q-a)^2 + b^2 by A8,COMPTRIG:3
    .= q^2 - 2*q*a + 1 by A4;
  then |.qc - r.| >= 0 & (|.qc - r.|)^2 > (q - 1)^2 by A7,COMPLEX1:46;
  hence thesis by SQUARE_1:48;
end;
