
theorem
  for V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V
  holds diagker f is linearly-closed
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V;
  set V1 = diagker f;
  thus for v,u be Element of V st v in V1 & u in V1 holds v + u in V1
  proof
    let v,u being Element of V;
    assume that
A1: v in V1 and
A2: u in V1;
A3: ex b be Vector of V st b = u & f.(b,b)= 0.F_Complex by A2;
    then |.f.(v,u).|^2 <= |. f.(v,v).| * 0 by Th46,COMPLFLD:57;
    then
A4: |.f.(v,u).| = 0 by XREAL_1:63;
    then 0 = |.(f.(v,u))*'.| by COMPLEX1:53
      .= |.f.(u,v).| by Def5;
    then
A5: f.(u,v) = 0.F_Complex by COMPLFLD:58;
    ex a be Vector of V st a = v & f.(a,a)= 0.F_Complex by A1;
    then
A6: f.(v+u,v+u) = 0.F_Complex + f.(v,u) +(f.(u,v)+0.F_Complex) by A3,
BILINEAR:28
      .= f.(v,u) +(f.(u,v)+0.F_Complex) by RLVECT_1:4
      .= f.(v,u) +f.(u,v) by RLVECT_1:4;
    f.(v,u) = 0.F_Complex by A4,COMPLFLD:58;
    then f.(v+u,v+u) = 0.F_Complex by A6,A5,RLVECT_1:4;
    hence thesis;
  end;
  let a be Element of F_Complex;
  let v be Element of V;
  assume v in V1;
  then
A7: ex w be Vector of V st w=v & f.(w,w)=0.F_Complex;
  f.(a*v,a*v) = a *f.(v,a*v) by BILINEAR:31
    .= a*(a*'* 0.F_Complex) by A7,Th27
    .=0.F_Complex;
  hence thesis;
end;
