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
  seq is Cauchy & (ex k st seq = seq1 ^\k) implies seq1 is Cauchy
proof
  assume that
A1: seq is Cauchy and
A2: ex k st seq = seq1 ^\k;
  consider k such that
A3: seq = seq1 ^\k by A2;
  let r;
  assume r > 0;
  then consider l1 such that
A4: for n, m st n >= l1 & m >= l1 holds dist((seq.n), (seq.m)) < r by A1;
  take l = l1 + k;
  let n, m;
  assume that
A5: n >= l and
A6: m >= l;
  consider m1 being Nat such that
A7: n = l1 + k + m1 by A5,NAT_1:10;
  reconsider m1 as Nat;
  n - k = l1 + m1 by A7;
  then consider l2 such that
A8: l2 = n - k;
  consider m2 being Nat such that
A9: m = l1 + k + m2 by A6,NAT_1:10;
  reconsider m2 as Nat;
  m - k = l1 + m2 by A9;
  then consider l3 such that
A10: l3 = m - k;
A11: now
    assume l2 < l1;
    then l2 + k < l1 + k by XREAL_1:6;
    hence contradiction by A5,A8;
  end;
A12: l2 + k = n by A8;
  now
    assume l3 < l1;
    then l3 + k < l1 + k by XREAL_1:6;
    hence contradiction by A6,A10;
  end;
  then dist((seq.l2), (seq.l3)) < r by A4,A11;
  then
A13: dist((seq1.n), (seq.l3)) < r by A3,A12,NAT_1:def 3;
  l3 + k = m by A10;
  hence thesis by A3,A13,NAT_1:def 3;
end;
