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
  for X being RealHilbertSpace, seq being sequence of X
   holds seq is absolutely_summable implies seq is
  summable
proof
 let X be RealHilbertSpace, seq be sequence of X;
  assume that
A1: seq is absolutely_summable;
A2: ||.seq.|| is summable by A1;
  now
    let r;
    assume r > 0;
    then consider k being Nat such that
A3: for m being Nat
   st m >= k holds |.Partial_Sums(||.seq.||).m - Partial_Sums
    (||.seq.||).k.| < r/2 by A2,SERIES_1:21,XREAL_1:215;
     reconsider k as Nat;
    take k;
    now
      let m, n;
A4:   ||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| <= |.Partial_Sums
      (||.seq.||).n - Partial_Sums(||.seq.||).m.| by Th39;
      assume m >= k & n >= k;
      then
      |.Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k.| < r /2 &
      |. Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.|| ).k.| < r/2 by A3;
      then
A5:   |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k.| + |.
      Partial_Sums(||.seq.||).m - Partial_Sums(||.seq.||).k.| < r/2 + r/2 by
XREAL_1:8;
      |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| = |.(
Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k) - (Partial_Sums(||.seq
      .||).m - Partial_Sums(||.seq.|| ).k).|;
      then |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| <= |.
Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).k.| + |.Partial_Sums(||.
      seq.||).m - Partial_Sums(||.seq.||).k.| by COMPLEX1:57;
      then |.Partial_Sums(||.seq.||).n - Partial_Sums(||.seq.||).m.| < r by A5
,XXREAL_0:2;
      hence ||.Partial_Sums(seq).n - Partial_Sums(seq).m.|| < r by A4,
XXREAL_0:2;
    end;
    hence for n, m st n >= k & m >= k holds ||.Partial_Sums(seq).n -
    Partial_Sums(seq).m.|| < r;
  end;
  hence thesis by Th10;
end;
