
theorem Th7:
  for s being FinSequence of REAL st s is nonnegative-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 nonnegative-yielding and
    A2: ex i being Nat st i in dom s & s.i <> 0;
  consider i being Nat such that
    A3: i in dom s and
    A4: s.i <> 0 by A2;
  set sr = s;
  A5: for j being Nat st j in dom sr holds 0 <= sr.j
  proof
    let j be Nat;
    assume j in dom sr;
    then sr.j in rng sr by FUNCT_1:3;
    hence 0 <= sr.j by A1, PARTFUN3:def 4;
  end;
  ex k be Nat st k in dom sr & 0 < sr.k
  proof
    take i;
    thus i in dom sr by A3;
    sr.i in rng sr by A3, FUNCT_1:3;
    hence 0 < sr.i by A1, A4, PARTFUN3:def 4;
  end;
  hence 0 < Sum s by A5, RVSUM_1:85;
end;
