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 Th9:
  seq is summable implies seq is convergent & lim seq = 0.X
proof
  set PSseq = Partial_Sums(seq);
  now
    let n;
    (PSseq).(n + 1) = (PSseq).n + seq.(n + 1) by BHSP_4:def 1
      .= (PSseq).n + (seq ^\1).n by NAT_1:def 3;
    hence (PSseq ^\1).n = (PSseq).n + (seq ^\1).n by NAT_1:def 3;
  end;
  then
A1: (PSseq ^\1) = PSseq + seq ^\1 by NORMSP_1:def 2;
  now
    let n be Element of NAT;
    thus (seq ^\1 + (PSseq - PSseq)).n = (seq ^\1).n + (PSseq - PSseq).n by
NORMSP_1:def 2
      .= (seq ^\1).n + (PSseq.n - PSseq.n) by NORMSP_1:def 3
      .= (seq ^\1).n + 09(X) by RLVECT_1:15
      .= (seq ^\1).n by RLVECT_1:4;
  end;
  then seq ^\1 + (PSseq - PSseq) = seq ^\1 by FUNCT_2:63;
  then
A2: seq ^\1 = PSseq ^\1 - PSseq by A1,CSSPACE:76;
  assume seq is summable;
  then
A3: PSseq is convergent;
  then
A4: PSseq ^\1 is convergent by CLVECT_2:90;
  then
A5: seq ^\1 is convergent by A3,A2,CLVECT_2:4;
  hence seq is convergent by CLVECT_2:91;
  lim(PSseq ^\1) = lim(PSseq) by A3,CLVECT_2:90;
  then lim(PSseq ^\1 - PSseq) = lim(PSseq) - lim(PSseq) by A3,A4,CLVECT_2:14
    .= 09(X) by RLVECT_1:15;
  hence thesis by A2,A5,CLVECT_2:84,91;
end;
