
theorem
  for s being FinSequence of REAL st s is nonpositive-yielding &
    ex i being Nat st i in dom s & s.i <> 0 holds
      Sum s < 0
proof
  let s be FinSequence of REAL;
  assume that
    A1: s is nonpositive-yielding and
    A2: ex i being Nat st i in dom s & s.i <> 0;
  reconsider D = dom s as non empty set by A2;
  rng s c= REAL;
  then reconsider sr = s as nonpositive-yielding Function of D, REAL
    by A1, FUNCT_2:2;
  reconsider ms = -s as FinSequence of REAL;
a3: ms = -sr;
  rng s c= COMPLEX by NUMBERS:11;
  then reconsider sc = s as Function of D, COMPLEX by FUNCT_2:2;
  A4: dom sc = dom ms by CFUNCT_1:5;
  ex i being Nat st i in dom ms & ms.i <> 0
  proof
    consider i being Nat such that
      A5: i in dom s and
      A6: s.i <> 0 by A2;
    take i;
    thus i in dom ms by A5, A4;
    assume ms.i = 0;
    then -(sr.i) = 0 by A5, RFUNCT_1:58;
    hence contradiction by A6;
  end;
  then Sum ms > 0 by a3, Th7;
  then - Sum s > 0 by RVSUM_1:88;
  hence Sum s < 0;
end;
