
theorem Th2:
  for z be Element of COMPLEX holds |.z.|+0*<i> = (z*'/(|.z.|+0*<i> ))*z
proof
  let z be Element of COMPLEX;
  per cases;
  suppose
A1: |.z.| = 0;
    then z = 0 by COMPLEX1:45;
    hence thesis by A1;
  end;
  suppose
A2: |.z.| <> 0;
A3: Im(z*z*') = 0 by COMPLEX1:40;
    |.z.| = |.z.|+0*<i>;
    then
A4: Re |.z.| = |.z.| & Im |.z.| = 0 by COMPLEX1:12;
A5: (z*'/|.z.|)*z = z*z*'/|.z.| & Re(z*z*') = (Re z)^2 + (Im z)^2 by
COMPLEX1:40,XCMPLX_1:74;
    then
A6: Im((z*'/|.z.|)*z) = (|.z.|*0 - ((Re z)^2+(Im z)^2)*0) / (|.z.| ^2 +0^2
    ) by A3,A4,COMPLEX1:24;
    Re((z*'/|.z.|)*z) = (((Re z)^2+(Im z)^2)*|.z.| + 0*0) / (|.z .|^2 +0^2
    ) by A5,A3,A4,COMPLEX1:24
      .= |.z*z.|*|.z.| / (|.z.|*|.z.|) by COMPLEX1:68
      .= |.z*z.| / |.z.| by A2,XCMPLX_1:91
      .= |.z.|*|.z.| / |.z.| by COMPLEX1:65
      .= |.z.| by A2,XCMPLX_1:89;
    hence thesis by A6,COMPLEX1:13;
  end;
end;
