
theorem Th6:
  for s being FinSequence of REAL st Sum s <> 0 holds
    ex i being Nat st i in dom s & s.i <> 0
proof
  let s be FinSequence of REAL;
  assume A1: Sum s <> 0;
  consider n being Nat such that A2: dom s = Seg n by FINSEQ_1:def 2;
  assume for i being Nat holds not i in dom s or s.i = 0;
  then for i being object st i in dom s holds s.i = 0;
  then A3: s = dom s --> 0 by FUNCOP_1:11;
  s = n |-> 0 by A3, A2, FINSEQ_2:def 2;
  hence contradiction by A1,RVSUM_1:81;
end;
