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_compared_to seq2 iff for r st r > 0 ex m st for n st n >= m
  holds ||.(seq1.n) - (seq2.n).|| < r
proof
  thus seq1 is_compared_to seq2 implies for r st r > 0 ex m st for n st n >= m
  holds ||.(seq1.n) - (seq2.n).|| < r
  proof
    assume
A1: seq1 is_compared_to seq2;
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A2: for n st n >= m1 holds dist((seq1.n), (seq2.n)) < r by A1;
    take m = m1;
    let n;
    assume n >= m;
    then dist((seq1.n), (seq2.n)) < r by A2;
    hence thesis by CSSPACE:def 16;
  end;
  ( for r st r > 0 ex m st for n st n >= m holds ||.(seq1.n) - (seq2.n).||
  < r ) implies seq1 is_compared_to seq2
  proof
    assume
A3: for r st r > 0 ex m st for n st n >= m holds ||.(seq1.n) - (seq2.n
    ).|| < r;
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A4: for n st n >= m1 holds ||.(seq1.n) - (seq2.n).|| < r by A3;
    take m = m1;
    let n;
    assume n >= m;
    then ||.(seq1.n) - (seq2.n).|| < r by A4;
    hence thesis by CSSPACE:def 16;
  end;
  hence thesis;
end;
