
theorem
  for a be Complex ex b be Complex st |.b.| = 1 &
  Re (b * a) = |.a.| & Im (b * a)= 0
proof
  let z be Complex;
  set r = |.z.|;
A1: r = 0 implies ex a be Element of COMPLEX st |.a.| =1 & Re (a * z) = r &
  Im (a * z)= 0
  proof
    assume
A2: r = 0;
    take 1r;
    thus thesis by A2,COMPLEX1:4,45,48;
  end;
  0 < r implies ex a be Complex st |.a.| =1 & Re (a * z) = r &
  Im (a * z)= 0
  proof
    assume
A3: 0 < r;
    take (Re z)/r+ (-Im z)/r*<i>;
    thus thesis by A3,Th2,COMPLEX1:44;
  end;
  hence thesis by A1,COMPLEX1:46;
end;
