reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

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 Partial_Sums(seq^\k) is convergent;
    then
A2: Partial_Sums(seq^\k)^\1 is convergent by CLVECT_2:90;
    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,CLVECT_2:1;
    then
    seq^\(k+1)=(seq^\k)^\1 & Partial_Sums(seq^\k^\1) is convergent by A2,
CLVECT_2:4,NAT_1:48;
    hence thesis by Def1;
  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;
