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;
