reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of 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 be Nat such that
A4: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < r/2 by A1;
  consider m2 be Nat 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 CSSPACE:51;
  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;
