
theorem Th53:
  for V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form
  of V holds diagker f = rightker f
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V;
  thus diagker f c= rightker 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.(w,a).|^2 <= |. f.(w,w).|*0 by A2,Th46,COMPLFLD:57;
      then |.f.(w,a).| = 0 by XREAL_1:63;
      hence f.(w,a)=0.F_Complex by COMPLFLD:58;
    end;
    hence thesis by A1;
  end;
  thus thesis by BILINEAR:41;
end;
