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
  seq1 is_compared_to seq2 iff for r st r > 0 ex m st for n st n >= m
  holds ||.(seq1.n) - (seq2.n).|| < r
proof
  thus seq1 is_compared_to seq2 implies for r st r > 0 ex m st for n st n >= m
  holds ||.(seq1.n) - (seq2.n).|| < r
  proof
    assume
A1: seq1 is_compared_to seq2;
    let r;
    assume r > 0;
    then consider m1 such that
A2: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < r by A1;
    take m = m1;
    let n;
    assume n >= m;
    then dist((seq1.n), (seq2.n)) < r by A2;
    hence thesis by BHSP_1:def 5;
  end;
  ( for r st r > 0 ex m st for n st n >= m holds ||.(seq1.n) - (seq2.n).||
  < r ) implies seq1 is_compared_to seq2
  proof
    assume
A3: for r st r > 0 ex m st for n st n >= m holds ||.(seq1.n) - (seq2.n
    ).|| < r;
    let r;
    assume r > 0;
    then consider m1 such that
A4: for n st n >= m1 holds ||.(seq1.n) - (seq2.n).|| < r by A3;
    take m = m1;
    let n;
    assume n >= m;
    then ||.(seq1.n) - (seq2.n).|| < r by A4;
    hence thesis by BHSP_1:def 5;
  end;
  hence thesis;
end;
