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 Cauchy & seq1 is_compared_to seq2 implies seq2 is Cauchy
proof
  assume that
A1: seq1 is Cauchy and
A2: seq1 is_compared_to seq2;
  let r;
  assume r > 0;
  then
A3: r/3 > 0 by XREAL_1:222;
  then consider m1 such that
A4: for n, m st n >= m1 & m >= m1 holds dist((seq1.n), (seq1.m)) < r/3
  by A1;
  consider m2 such that
A5: for n st n >= m2 holds dist((seq1.n), (seq2.n)) < r/3 by A2,A3;
  take k = m1 + m2;
  let n, m such that
A6: n >= k and
A7: m >= k;
  m1 + m2 >= m1 by NAT_1:12;
  then n >= m1 & m >= m1 by A6,A7,XXREAL_0:2;
  then
A8: dist((seq1.n), (seq1.m)) < r/3 by A4;
A9: dist((seq2.n), (seq1.m)) <= dist((seq2.n), (seq1.n)) + dist((seq1.n), (
  seq1.m)) by BHSP_1:35;
A10: k >= m2 by NAT_1:12;
  then n >= m2 by A6,XXREAL_0:2;
  then dist((seq1.n), (seq2.n)) < r/3 by A5;
  then
  dist((seq2.n), (seq1.n)) + dist((seq1.n), (seq1.m)) < r/3 + r/3 by A8,
XREAL_1:8;
  then
A11: dist((seq2.n), (seq1.m)) < r/3 + r/3 by A9,XXREAL_0:2;
A12: dist((seq2.n), (seq2.m)) <= dist((seq2.n), (seq1.m)) + dist((seq1.m), (
  seq2.m)) by BHSP_1:35;
  m >= m2 by A7,A10,XXREAL_0:2;
  then dist((seq1.m), (seq2.m)) < r/3 by A5;
  then dist((seq2.n), (seq1.m)) + dist((seq1.m), (seq2.m)) < r/3 + r/3 + r/3
  by A11,XREAL_1:8;
  hence thesis by A12,XXREAL_0:2;
end;
