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 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 Def1
      .= (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;
  seq ^\1 + (PSseq - PSseq) = seq ^\1
  proof
    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;
  end;
  then
A2: seq ^\1 = PSseq ^\1 - PSseq by A1,BHSP_1:61;
  assume seq is summable;
  then
A3: PSseq is convergent;
A4: seq ^\1 is convergent by A3,A2,BHSP_2:4;
  hence seq is convergent by BHSP_3:37;
  lim(PSseq ^\1) = lim(PSseq) by A3,BHSP_3:36;
  then lim(PSseq ^\1 - PSseq) = lim(PSseq) - lim(PSseq) by A3,BHSP_2:14
    .= 09(X) by RLVECT_1:15;
  hence thesis by A2,A4,BHSP_3:28,37;
end;
