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_compared_to seq2 implies (seq1 ^\k) is_compared_to (seq2 ^\k)
proof
  assume
A1: seq1 is_compared_to seq2;
  let r;
  assume r > 0;
  then consider m1 such that
A2: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < r by A1;
  take m = m1;
  let n such that
A3: n >= m;
  n + k >= n by NAT_1:11;
  then n + k >= m by A3,XXREAL_0:2;
  then dist((seq1.(n + k)), (seq2.(n + k))) < r by A2;
  then dist((seq1 ^\k).n, (seq2.(n + k))) < r by NAT_1:def 3;
  hence thesis by NAT_1:def 3;
end;
