reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th13:
  (ex n st s^\n is summable) implies s is summable
proof
  given n such that
A1: s^\n is summable;
  reconsider PS = Partial_Sums(s).n as Element of REAL;
  set s1 = seq_const Partial_Sums(s).n;
  for k holds (Partial_Sums(s)^\(n+1)).k = Partial_Sums(s^\(n+1)).k + s1.k
  proof
    defpred X[Nat] means
  (Partial_Sums(s)^\(n+1)).$1 = Partial_Sums
    (s^\(n+1)).$1 + s1.$1;
A2: Partial_Sums(s^\(n+1)).0 = (s^\(n+1)).0 by Def1
      .= s.(0+(n+1)) by NAT_1:def 3
      .= s.(n+1);
A3: for k st X[k] holds X[k+1]
    proof
      let k;
      assume
A4:   (Partial_Sums(s)^\(n+1)).k = Partial_Sums(s^\(n+1)).k + s1.k;
      Partial_Sums(s^\(n+1)).(k+1) + s1.(k+1) = s1.(k+1) + (Partial_Sums(
      s^\(n+1)).k + (s^\(n+1)).(k+1)) by Def1
        .= s1.(k+1) + Partial_Sums(s^\(n+1)).k + (s^\(n+1)).(k+1)
        .= Partial_Sums(s).n + Partial_Sums(s^\(n+1)).k + (s^\(n+1)).(k+1)
                  by SEQ_1:57
        .= (Partial_Sums(s)^\(n+1)).k + (s^\(n+1)).(k+1) by A4,SEQ_1:57
        .= Partial_Sums(s).(k+(n+1)) + (s^\(n+1)).(k+1) by NAT_1:def 3
        .= Partial_Sums(s).(k+(n+1)) + s.(k+1+(n+1)) by NAT_1:def 3
        .= Partial_Sums(s).(k+(n+1)+1) by Def1
        .= Partial_Sums(s).(k+1+(n+1))
        .= (Partial_Sums(s)^\(n+1)).(k+1) by NAT_1:def 3;
      hence thesis;
    end;
    (Partial_Sums(s)^\(n+1)).0 = Partial_Sums(s).(0+(n+1)) by NAT_1:def 3
      .= s.(n+1) + Partial_Sums(s).n by Def1
      .= Partial_Sums(s^\(n+1)).0 + s1.0 by A2,SEQ_1:57;
    then
A5: X[0];
    thus for k holds X[k] from NAT_1:sch 2(A5,A3);
  end;
  then
A6: Partial_Sums(s)^\(n+1) = Partial_Sums(s^\(n+1)) + s1 by SEQ_1:7
    .= Partial_Sums((s^\n)^\1) + s1 by NAT_1:48;
  s^\n^\1 is summable by A1,Th12;
  then Partial_Sums(s^\n^\1) is convergent;
  then Partial_Sums(s)^\(n+1) is convergent by A6;
  then Partial_Sums(s) is convergent by SEQ_4:21;
  hence thesis;
end;
