reserve c, c1, c2, d, d1, d2, e, y for Real,
  k, n, m, N, n1, N0, N1, N2, N3, M for Element of NAT,
  x for set;

theorem :: (Part 1, O(nlogn) c O(n^(1+e))), slightly generalized
  for e being Real, f being Real_Sequence st 0 < e & (for n st n > 0
holds f.n = n*log(2,n)) holds ex s being eventually-positive Real_Sequence st s
  = f & Big_Oh(s) c= Big_Oh(seq_n^(1+e)) & not Big_Oh(s) = Big_Oh(seq_n^(1+e))
proof
  set seq = seq_logn;
  let e be Real, f be Real_Sequence such that
A1: 0 < e and
A2: for n st n > 0 holds f.n = n*log(2,n);
  set seq1 = seq_n^(e);
  set p = seq /" seq1;
A3: lim p = 0 by A1,Lm11;
  f is eventually-positive
  proof
    take 2;
    let n be Nat;
A4:  n in NAT by ORDINAL1:def 12;
    assume
A5: n >= 2;
    then log(2,n) >= log(2,2) by PRE_FF:10;
    then log(2,n) >= 1 by POWER:52;
    then n*log(2,n) > n*0 by A5,XREAL_1:68;
    hence thesis by A2,A5,A4;
  end;
  then reconsider f as eventually-positive Real_Sequence;
  set g = seq_n^(1+e);
  set h = f /" g;
A6: for n st n >= 1 holds h.n = p.n
  proof
    let n;
    assume
A7: n >= 1;
    h.n = f.n / g.n by Lm4
      .= n*log(2,n) / g.n by A2,A7
      .= n*log(2,n) / (n to_power (1+e)) by A7,Def3
      .= (n to_power 1)*log(2,n) / (n to_power (1+e)) by POWER:25
      .= (n to_power 1)*log(2,n) * (n to_power (1+e))"
      .= log(2,n)*((n to_power 1)*(n to_power (1+e))")
      .= log(2,n)*((n to_power 1)/(n to_power (1+e)))
      .= log(2,n)*(n to_power (1-(1+e))) by A7,POWER:29
      .= log(2,n)*(n to_power (1+(-1+-e)))
      .= log(2,n)*(1/(n to_power e)) by A7,POWER:28
      .= log(2,n)/(n to_power e)
      .= seq.n / (n to_power e) by A7,Def2
      .= seq.n / seq1.n by A7,Def3
      .= p.n by Lm4;
    hence thesis;
  end;
A8: p is convergent by A1,Lm11;
  then
A9: lim h = 0 by A3,A6,Lm22;
A10: h is convergent by A8,A3,A6,Lm22;
  then not g in Big_Oh(f) by A9,ASYMPT_0:16;
  then
A11: not f in Big_Omega(g) by ASYMPT_0:19;
  take f;
  f in Big_Oh(g) by A10,A9,ASYMPT_0:16;
  hence thesis by A11,Th4;
end;
