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 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 be Nat such that
A4: for n, m st n >= m1 & m >= m1 holds dist((seq1.n), (seq1.m)) < r/3
  by A1;
  consider m2 be Nat 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 CSSPACE:51;
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 CSSPACE:51;
  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;
