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
  for e being Real, g being Real_Sequence st e < 1 & (for n st n > 1
  holds g.n = (n to_power 2)/log(2,n)) holds ex s being eventually-positive
Real_Sequence st s = g & Big_Oh(seq_n^(1+e)) c= Big_Oh(s) & not Big_Oh(seq_n^(1
  +e)) = Big_Oh(s)
proof
  set seq = seq_logn;
  let e be Real, g be Real_Sequence such that
A1: e < 1 and
A2: for n st n > 1 holds g.n = (n to_power 2)/log(2,n);
  set seq1 = seq_n^(1-e);
  set p = seq /" seq1;
  set f = seq_n^(1+e);
  set h = f /" g;
  g is eventually-positive
  proof
    take 2;
    let n be Nat;
A3:  n in NAT by ORDINAL1:def 12;
    assume
A4: n >= 2;
    then log(2,n) >= log(2,2) by PRE_FF:10;
    then
A5: log(2,n) >= 1 by POWER:52;
    n > 1 by A4,XXREAL_0:2;
    then
A6: g.n = (n to_power 2)/log(2,n) by A2,A3
      .= (n to_power 2)*(log(2,n))";
    n to_power 2 > 0 by A4,POWER:34;
    then (n to_power 2)*(log(2,n))" > (n to_power 2)*0 by A5,XREAL_1:68;
    hence thesis by A6;
  end;
  then reconsider g as eventually-positive Real_Sequence;
A7: (1+e)-2 = e-1;
A8: for n st n >= 2 holds h.n = p.n
  proof
    let n;
    assume
A9: n >= 2;
    then
A10: n > 1 by XXREAL_0:2;
    h.n = f.n / g.n by Lm4
      .= (n to_power (1+e)) / g.n by A9,Def3
      .= (n to_power (1+e)) / ((n to_power 2) / log(2,n)) by A2,A10
      .= (n to_power (1+e)) * ((n to_power 2) / log(2,n))"
      .= (n to_power (1+e)) * (log(2,n) / (n to_power 2)) by XCMPLX_1:213
      .= (n to_power (1+e)) * (log(2,n) * (n to_power 2)")
      .= ((n to_power (1+e)) * (n to_power 2)") * log(2,n)
      .= ((n to_power (1+e)) / (n to_power 2)) * log(2,n)
      .= (n to_power -(1-e)) * log(2,n) by A7,A9,POWER:29
      .= log(2,n) * (1 / (n to_power (1-e))) by A9,POWER:28
      .= log(2,n) / (n to_power (1-e))
      .= seq.n / (n to_power (1-e)) by A9,Def2
      .= seq.n / seq1.n by A9,Def3
      .= p.n by Lm4;
    hence thesis;
  end;
  take g;
  0+e < 1 by A1;
  then
A11: 0 < 1-e by XREAL_1:20;
  then
A12: p is convergent by Lm11;
A13: lim p = 0 by A11,Lm11;
  then
A14: lim h = 0 by A12,A8,Lm22;
A15: h is convergent by A12,A13,A8,Lm22;
  then not g in Big_Oh(f) by A14,ASYMPT_0:16;
  then
A16: not f in Big_Omega(g) by ASYMPT_0:19;
  f in Big_Oh(g) by A15,A14,ASYMPT_0:16;
  hence thesis by A16,Th4;
end;
