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
  ( 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 CSSPACE:50;
  hence thesis by A2;
end;
