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

theorem Th12:
  s is summable implies for n holds s^\n is summable
proof
  defpred X[Nat] means s^\$1 is summable;
A1: for n st X[n] holds X[n+1]
  proof
    let n;
    set s1 = seq_const((s^\n).0);
    for k holds s1.k = (s^\n).0 by SEQ_1:57;
    then
A2: Partial_Sums(s^\n^\1) = (Partial_Sums(s^\n)^\1) - s1 by Th11;
    assume s^\n is summable;
    then Partial_Sums(s^\n) is convergent;
    then s^\(n+1)=(s^\n)^\1 & Partial_Sums(s^\n^\1) is convergent by A2,
NAT_1:48;
    hence thesis by Def2;
  end;
  assume s is summable;
  then
A3: X[0] by NAT_1:47;
  thus for n holds X[n] from NAT_1:sch 2(A3,A1);
end;
