reserve X for RealUnitarySpace,
  x, g, g1, h for Point of X,
  a, p, r, M, M1, M2 for Real,
  seq, seq1, seq2, seq3 for sequence of X,
  Nseq for increasing sequence of NAT,

  k, l, l1, l2, l3, n, m, m1, m2 for Nat;

theorem
  seq is Cauchy iff for r st r > 0 ex k st for n, m st n >= k & m >= k
  holds ||.(seq.n) - (seq.m).|| < r
proof
  thus seq is Cauchy implies for r st r > 0 ex k st for n, m st n >= k & m >=
  k holds ||.(seq.n) - (seq.m).|| < r
  proof
    assume
A1: seq is Cauchy;
    let r;
    assume r > 0;
    then consider l such that
A2: for n, m st n >= l & m >= l holds dist((seq.n), (seq.m)) < r by A1;
    take k = l;
    let n, m;
    assume n >= k & m >= k;
    then dist((seq.n), (seq.m)) < r by A2;
    hence thesis by BHSP_1:def 5;
  end;
  ( for r st r > 0 ex k st for n, m st ( n >= k & m >= k ) holds ||.(seq.n
  ) - (seq.m).|| < r ) implies seq is Cauchy
  proof
    assume
A3: for r st r > 0 ex k st for n, m st n >= k & m >= k holds ||.(seq.
    n) - (seq.m).|| < r;
    let r;
    assume r > 0;
    then consider l such that
A4: for n, m st n >= l & m >= l holds ||.(seq.n) - (seq.m).|| < r by A3;
    take k = l;
    let n, m;
    assume n >= k & m >= k;
    then ||.(seq.n) - (seq.m).|| < r by A4;
    hence thesis by BHSP_1:def 5;
  end;
  hence thesis;
end;
