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
  (ex k st seq^\k is summable) implies seq is summable
proof
  given k such that
A1: seq^\k is summable;
  seq^\k^\1 is summable by A1,Th13;
  then
A2: Partial_Sums(seq^\k^\1) is convergent;
  reconsider seq1 = NAT --> Partial_Sums(seq).k as sequence of X;
  defpred P[Nat] means
(Partial_Sums(seq)^\(k+1)).$1 = Partial_Sums
  (seq^\(k+1)).$1 + seq1.$1;
A3: Partial_Sums(seq^\(k+1)).0 = (seq^\(k+1)).0 by Def1
    .= seq.(0+(k+1)) by NAT_1:def 3
    .= seq.(k+1);
A4: now
    let m;
A5:  m in NAT by ORDINAL1:def 12;
    assume
A6: P[m];
    Partial_Sums(seq^\(k+1)).(m+1) + seq1.(m+1) = seq1.(m+1) + (
    Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1)) by Def1
      .= seq1.(m+1) + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+1) by
RLVECT_1:def 3
      .= Partial_Sums(seq).k + Partial_Sums(seq^\(k+1)).m + (seq^\(k+1)).(m+
    1)
      .= (Partial_Sums(seq)^\(k+1)).m + (seq^\(k+1)).(m+1) by A6,FUNCOP_1:7,A5
      .= Partial_Sums(seq).(m+(k+1)) + (seq^\(k+1)).(m+1) by NAT_1:def 3
      .= Partial_Sums(seq).(m+(k+1)) + seq.(m+1+(k+1)) by NAT_1:def 3
      .= Partial_Sums(seq).(m+(k+1)+1) by Def1
      .= Partial_Sums(seq).(m+1+(k+1))
      .= (Partial_Sums(seq)^\(k+1)).(m+1) by NAT_1:def 3;
    hence P[m+1];
  end;
  (Partial_Sums(seq)^\(k+1)).0 = Partial_Sums(seq).(0+(k+1)) by NAT_1:def 3
    .= Partial_Sums(seq).k + seq.(k+1) by Def1
    .= Partial_Sums(seq^\(k+1)).0 + seq1.0 by A3;
  then
A7: P[0];
  for m holds P[m] from NAT_1:sch 2(A7,A4);
  then
A8: Partial_Sums(seq)^\(k+1) = Partial_Sums(seq^\(k+1)) + seq1 by
NORMSP_1:def 2
    .= Partial_Sums((seq^\k)^\1) + seq1 by BHSP_3:31;
  Partial_Sums(seq)^\(k+1) is convergent by A2,A8,BHSP_2:3;
  hence thesis by BHSP_3:37;
end;
