
theorem Th2:
  for a be Complex st a <> 0c holds |.(Re a)/|.a.|+ (-Im
a)/|.a.|*<i>.| = 1 & Re (((Re a)/|.a.|+ (-Im a)/|.a.|*<i>) * a) = |.a.| & Im ((
  (Re a)/|.a.|+ (-Im a)/|.a.|*<i>) * a)= 0
proof
  let z be Complex;
  set r = |.z.|, a = (Re z)/r+ (-Im z)/r*<i>;
  assume z <> 0c;
  then
A1: 0 < r by COMPLEX1:47;
  |.z*z.| = (Re z)^2 + (Im z)^2 by COMPLEX1:68;
  then 0 <= (Re z)^2 + (Im z)^2 by COMPLEX1:46;
  then
A2: r^2 = (Re z)^2 + (Im z)^2 by SQUARE_1:def 2;
A3: Re a = (Re z)/r & Im a = (-Im z)/r by COMPLEX1:12;
  then (Re a)^2 + (Im a)^2 = (Re z)^2/r^2 + ((-Im z)/r)^2 by XCMPLX_1:76
    .= (Re z)^2/r^2 + (-Im z)^2/r^2 by XCMPLX_1:76
    .= ((Re z)^2 + (Im z)^2)/r^2 by XCMPLX_1:62
    .= 1^2 by A1,A2,XCMPLX_1:60;
  hence |.a.| = 1;
  thus Re (a*z) = (Re z)/r * Re z - (-Im z)/r * Im z by A3,COMPLEX1:9
    .= ((Re z)*(Re z))/r - (-Im z)/r * Im z by XCMPLX_1:74
    .= (Re z)^2/r - ((-Im z)* Im z )/r by XCMPLX_1:74
    .= ((Re z)^2 - -(Im z * Im z) )/r by XCMPLX_1:120
    .= r by A1,A2,XCMPLX_1:89;
  thus Im(a*z) = (Re z)/r * Im z + Re z * ((-Im z)/r) by A3,COMPLEX1:9
    .= (Re z) * (Im z) /r + Re z * ((-Im z)/r) by XCMPLX_1:74
    .= (Re z) * (Im z) /r + (-Im z)* (Re z) /r by XCMPLX_1:74
    .= ((Re z) * (Im z) + -(Im z)* (Re z)) /r by XCMPLX_1:62
    .=0;
end;
