reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem Th13:
  seq is summable implies for k holds seq^\k is summable
proof
  defpred P[Nat] means seq^\($1) is summable;
A1: for k st P[k] holds P[k+1]
  proof
    let k;
    reconsider seq1 = NAT --> (seq^\k).0 as sequence of X;
    assume seq^\k is summable;
    then
A2:  Partial_Sums(seq^\k) is convergent;
    for m holds seq1.m = (seq^\k).0
      by ORDINAL1:def 12,FUNCOP_1:7;
    then
    seq1 is convergent & Partial_Sums(seq^\k^\1) = (Partial_Sums(seq^\k)^\
    1) - seq1 by Th12;
    then
    seq^\(k+1)=(seq^\k)^\1 & Partial_Sums(seq^\k^\1) is convergent
     by A2,BHSP_2:4,BHSP_3:31;
    hence thesis by Def2;
  end;
  assume seq is summable;
  then
A3: P[0] by NAT_1:47;
  thus for k holds P[k] from NAT_1:sch 2(A3,A1);
end;
