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
  ( ex k st for n st n >= k holds seq1.n = seq2.n ) implies seq1
  is_compared_to seq2
proof
  assume ex k st for n st n >= k holds seq1.n = seq2.n;
  then consider m such that
A1: for n st n >= m holds seq1.n = seq2.n;
  let r such that
A2: r > 0;
  take k = m;
  let n;
  assume n >= k;
  then dist((seq1.n), (seq2.n)) = dist((seq1.n), (seq1.n)) by A1
    .= 0 by BHSP_1:34;
  hence thesis by A2;
end;
