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 summable iff for r st r > 0 ex k st for
  n, m st n >= k & m >= k holds ||.Sum(seq, n, m).|| < r
proof
 let X be RealHilbertSpace, seq be sequence of X;
  thus seq is summable implies for r st r > 0 ex k st for n, m st n >= k & m
  >= k holds ||.Sum(seq, n, m).|| < r
  proof
    assume
A1: seq is summable;
    now
      let r;
      assume r > 0;
      then consider k such that
A2:   for n, m st n >= k & m >= k holds ||.Sum(seq, n) - Sum(seq, m)
      .|| < r by A1,Th23;
      take k;
      let n, m;
      assume n >= k & m >= k;
      hence ||.Sum(seq, n, m).|| < r by A2;
    end;
    hence thesis;
  end;
  thus ( for r st r > 0 ex k st for n, m st n >= k & m >= k holds ||.Sum(seq,
  n, m).|| < r ) implies seq is summable
  proof
    assume
A3: for r st r > 0 ex k st for n, m st n >= k & m >= k holds ||.Sum(
    seq, n, m).|| < r;
    now
      let r;
      assume r > 0;
      then consider k such that
A4:   for n, m st n >= k & m >= k holds ||.Sum(seq, n, m).|| < r by A3;
      take k;
      let n, m;
      assume n >= k & m >= k;
      then ||.Sum(seq, n, m).|| < r by A4;
      hence ||.Sum(seq, n) - Sum(seq, m).|| < r;
    end;
    hence thesis by Th23;
  end;
end;
