
theorem Th52:
  for V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form
  of V holds diagker f = leftker f
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V;
  thus diagker f c= leftker f
  proof
    let x be object;
    assume x in diagker f;
    then consider a be Vector of V such that
A1: a=x and
A2: f.(a,a) = 0. F_Complex;
    now
      let w be Vector of V;
      |.f.(a,w).|^2 <= 0*|. f.(w,w).| by A2,Th46,COMPLFLD:57;
      then |.f.(a,w).| = 0 by XREAL_1:63;
      hence f.(a,w)=0.F_Complex by COMPLFLD:58;
    end;
    hence thesis by A1;
  end;
  let x be object;
  assume x in leftker f;
  then consider a be Vector of V such that
A3: a=x and
A4: for w be Vector of V holds f.(a,w) = 0.F_Complex;
  f.(a,a) = 0.F_Complex by A4;
  hence thesis by A3;
end;
