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 & seq2 is_compared_to seq3 implies seq1
  is_compared_to seq3
proof
  assume that
A1: seq1 is_compared_to seq2 and
A2: seq2 is_compared_to seq3;
  let r;
  assume r > 0;
  then
A3: r/2 > 0 by XREAL_1:215;
  then consider m1 such that
A4: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < r/2 by A1;
  consider m2 such that
A5: for n st n >= m2 holds dist((seq2.n), (seq3.n)) < r/2 by A2,A3;
  take m = m1 + m2;
  let n such that
A6: n >= m;
  m >= m2 by NAT_1:12;
  then n >= m2 by A6,XXREAL_0:2;
  then
A7: dist((seq2.n), (seq3.n)) < r/2 by A5;
A8: dist((seq1.n), (seq3.n)) <= dist((seq1.n), (seq2.n)) + dist((seq2.n), (
  seq3.n)) by BHSP_1:35;
  m1 + m2 >= m1 by NAT_1:12;
  then n >= m1 by A6,XXREAL_0:2;
  then dist((seq1.n), (seq2.n)) < r/2 by A4;
  then dist((seq1.n), (seq2.n)) + dist((seq2.n), (seq3.n)) < r/2 + r/2 by A7,
XREAL_1:8;
  hence thesis by A8,XXREAL_0:2;
end;
