
theorem Th36:
  for a be NAT-valued Real_Sequence,b be non trivial Nat, n be Nat
  st a is eventually-non-zero & rng a c= b holds
  Sum(Liouville_seq (a,b) ^\ (n+1)) > 0
  proof
    let a be NAT-valued Real_Sequence, b be non trivial Nat, n be Nat;
    assume
A1: a is eventually-non-zero & rng a c= b;
    set LS = Liouville_seq (a,b) ^\ (n + 1);
A2: for i be Nat holds 0 <= LS.i
    proof
      let i be Nat;
      LS.i = Liouville_seq (a,b).(n+1+i) by NAT_1:def 3;
      hence thesis by Th33;
    end;
    ex i be Nat st i in dom LS & 0 < LS.i
    proof
      consider j be Nat such that
A3:   n + 1 <= j & a.j <> 0 by A1;
      j - (n + 1) in NAT by A3,INT_1:5; then
      reconsider i = j - (n + 1) as Nat;
      take i;
A4:   n + 1 + i = j;
A5:   dom LS = NAT by FUNCT_2:def 1;
      LS.i = Liouville_seq (a,b).j by NAT_1:def 3,A4
        .= (a.(n + 1 + i)) / (b to_power (j!)) by DefLio;
      hence thesis by A5,A3,ORDINAL1:def 12;
    end; then
    consider k be Nat such that
A6: k in dom LS & LS.k > 0;
    Liouville_seq (a,b) is summable by Th31,A1,NAT_2:29; then
    LS is summable by SERIES_1:12;
    hence thesis by A6,RSSPACE2:3,A2;
  end;
