
theorem Th66:
  for V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form
  of V holds leftker ScalarForm(f) = leftker Q*Form(f)
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V;
  set vq = VectQuot(V,LKer(f)), vr = VectQuot(V,RKer(f*')), qf=Q*Form(f), qhf=
  ScalarForm(f), K = F_Complex;
  thus leftker qhf c= leftker qf
  proof
    let x be object;
    assume x in leftker qhf;
    then consider A be Vector of vq such that
A1: x= A and
A2: for B be Vector of vq holds qhf.(A,B)=0.K;
    now
      let B be Vector of vr;
      reconsider w=B as Vector of vq by Th56;
      thus qf.(A,B) = qhf.(A,w) .= 0.K by A2;
    end;
    hence thesis by A1;
  end;
  let x be object;
  assume x in leftker qf;
  then consider A be Vector of vq such that
A3: x= A and
A4: for B be Vector of vr holds qf.(A,B)=0.K;
  now
    let B be Vector of vq;
    reconsider w=B as Vector of vr by Th56;
    thus qhf.(A,B) = qf.(A,w) .= 0.K by A4;
  end;
  hence thesis by A3;
end;
