reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem
  for f being real-valued FinSequence st |.f.| <> 0
  ex i being Nat st i in dom f & f.i <> 0
  proof
    let f be real-valued FinSequence;
    assume |.f.| <> 0;
    then consider i being Element of NAT such that
A1: i in dom sqr f and
A2: (sqr f).i <> 0 by PRVECT_2:3,SQUARE_1:17;
    take i;
    thus i in dom f by A1,VALUED_1:11;
    (sqr f).i = (f.i)^2 by VALUED_1:11;
    hence thesis by A2;
  end;
