
theorem Th8:
  for a be Element of F_Complex ex b be Element of F_Complex st |.b
  .| = 1 & Re (b * a) = |.a.| & Im (b * a)= 0
proof
  let z be Element of F_Complex;
  set r = |.z.|;
A1: r = 0 implies ex a be Element of F_Complex st |.a.| = 1 & Re (a * z) = r
  & Im (a * z) = 0
  proof
    assume
A2: r = 0;
    take a=1.F_Complex;
    thus |.a.| = 1 by COMPLFLD:60;
    z = 0.F_Complex by A2,COMPLFLD:58
      .= 0 by COMPLFLD:def 1;
    hence thesis by A2,Lm1,COMPLEX1:12;
  end;
  0 < r implies ex a be Element of F_Complex st |.a.| =1 & Re (a * z) = r
  & Im (a * z)= 0
  proof
    assume
A3: 0 < r;
    take [**(Re z)/r, (-Im z)/r**];
    thus thesis by A3,Th7,COMPLFLD:57;
  end;
  hence thesis by A1,COMPLEX1:46;
end;
