
theorem Th58:
  for V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form
of V, v be Vector of V st Re (f.(v,v))= 0 & (f is non degenerated-on-left or f
  is non degenerated-on-right) holds v=0.V
proof
  let V be VectSp of F_Complex, f be diagReR+0valued hermitan-Form of V, v be
  Vector of V;
  assume that
A1: Re (f.(v,v))= 0 and
A2: f is non degenerated-on-left or f is non degenerated-on-right;
  f.(v,v) = 0.F_Complex by A1,Th57;
  then
A3: v in {w where w is Vector of V: f.(w,w)=0.F_Complex};
  per cases by A2;
  suppose
    f is non degenerated-on-left;
    then {0.V} = diagker f by Th52;
    hence thesis by A3,TARSKI:def 1;
  end;
  suppose
    f is non degenerated-on-right;
    then {0.V} = diagker f by Th53;
    hence thesis by A3,TARSKI:def 1;
  end;
end;
