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 & seq2 is Cauchy implies seq1 - seq2 is Cauchy
proof
  assume that
A1: seq1 is Cauchy and
A2: seq2 is Cauchy;
  let r;
  assume r > 0;
  then
A3: r/2 > 0 by XREAL_1:215;
  then consider m1 be Nat such that
A4: for n, m st n >= m1 & m >= m1 holds dist((seq1.n), (seq1.m)) < r/2
  by A1;
  consider m2 be Nat such that
A5: for n, m st n >= m2 & m >= m2 holds dist((seq2.n), (seq2.m)) < r/2
  by A2,A3;
  take k = m1 + m2;
  let n, m such that
A6: n >= k & m >= k;
  k >= m2 by NAT_1:12;
  then n >= m2 & m >= m2 by A6,XXREAL_0:2;
  then
A7: dist((seq2.n), (seq2.m)) < r/2 by A5;
  dist((seq1 - seq2).n, (seq1 - seq2).m) = dist((seq1.n) - (seq2.n), (
  seq1 - seq2).m) by NORMSP_1:def 3
    .= dist((seq1.n) - (seq2.n), (seq1.m) - (seq2.m)) by NORMSP_1:def 3;
  then
A8: dist((seq1 - seq2).n, (seq1 - seq2).m) <= dist((seq1.n), (seq1.m)) +
  dist((seq2.n), (seq2.m)) by CSSPACE:57;
  m1 + m2 >= m1 by NAT_1:12;
  then n >= m1 & m >= m1 by A6,XXREAL_0:2;
  then dist((seq1.n), (seq1.m)) < r/2 by A4;
  then dist((seq1.n), (seq1.m)) + dist((seq2.n), (seq2.m)) < r/2 + r/2 by A7,
XREAL_1:8;
  hence thesis by A8,XXREAL_0:2;
end;
