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 f being Real_Sequence st (for n st n > 1 holds f.n = (n to_power 2
  )/log(2,n)) holds ex s being eventually-positive Real_Sequence st s = f &
  Big_Oh(s) c= Big_Oh(seq_n^(8)) & not Big_Oh(s) = Big_Oh(seq_n^(8))
proof
  set g = seq_n^(8);
  let f be Real_Sequence such that
A1: for n st n > 1 holds f.n = (n to_power 2)/log(2,n);
A2: f 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: f.n = (n to_power 2)/log(2,n) by A1,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;
  set h = f/"g;
  reconsider f as eventually-positive Real_Sequence by A2;
A7: now
A8: log(2,3) > log(2,2) by POWER:57;
    let p be Real;
    assume
A9: p > 0;
A10: [/(p to_power -(1/6))\] >= (p to_power -(1/6)) by INT_1:def 7;
    reconsider p1 = p as Real;
    set N = max(3,[/(p1 to_power -(1/6))\]);
A11: N >= 3 by XXREAL_0:25;
A12: N is Integer by XXREAL_0:16;
A13: N >= [/(p to_power -(1/6))\] by XXREAL_0:25;
    N in NAT by A11,A12,INT_1:3;
    then reconsider N as Nat;
    take N;
    let n be Nat;
A14:  n in NAT by ORDINAL1:def 12;
    assume
A15: n >= N;
    then
A16: n >= 3 by A11,XXREAL_0:2;
    then
A17: n > 1 by XXREAL_0:2;
A18: h.n = f.n/g.n by Lm4
      .= ((n to_power 2)/log(2,n)) / g.n by A1,A17,A14
      .= ((n to_power 2)/log(2,n)) / (n to_power 8) by A11,A15,Def3,A14
      .= ((n to_power 2)*(log(2,n))") / (n to_power 8)
      .= ((log(2,n))"*(n to_power 2)) * (n to_power 8)"
      .= (log(2,n))"*((n to_power 2)*(n to_power 8)")
      .= (log(2,n))"*((n to_power 2)/(n to_power 8))
      .= (log(2,n))"*(n to_power (2-8)) by A11,A15,POWER:29
      .= (log(2,n))"*(n to_power -6)
      .= (log(2,n))"*(1/(n to_power 6)) by A11,A15,POWER:28
      .= (1/(n to_power 6))*(1/log(2,n))
      .= 1/((n to_power 6)*log(2,n)) by XCMPLX_1:102;
    n >= [/(p to_power -(1/6))\] by A13,A15,XXREAL_0:2;
    then
A19: n >= (p to_power -(1/6)) by A10,XXREAL_0:2;
    (p1 to_power -(1/6)) > 0 by A9,POWER:34;
    then n to_power 6 >= (p to_power -(1/6)) to_power 6 by A19,Lm6;
    then
A20: n to_power 6 >= p1 to_power ((-(1/6))*6) by A9,POWER:33;
    p1 to_power -1 > 0 by A9,POWER:34;
    then 1/(n to_power 6) <= 1/(p to_power -1) by A20,XREAL_1:85;
    then 1/(n to_power 6) <= 1/(1/(p1 to_power 1)) by A9,POWER:28;
    then
A21: 1/(n to_power 6) <= p by POWER:25;
    log(2,n) >= log(2,3) by A16,PRE_FF:10;
    then log(2,n) > log(2,2) by A8,XXREAL_0:2;
    then
A22: log(2,n) > 1 by POWER:52;
A23: n to_power 6 > 0 by A11,A15,POWER:34;
    then (n to_power 6)*1 < (n to_power 6)*log(2,n) by A22,XREAL_1:68;
    then h.n < 1/(n to_power 6) by A23,A18,XREAL_1:88;
    then h.n < p by A21,XXREAL_0:2;
    hence |.h.n-0.| < p by A22,A18,ABSVALUE:def 1;
  end;
  then
A24: h is convergent by SEQ_2:def 6;
  then
A25: lim h = 0 by A7,SEQ_2:def 7;
  then not g in Big_Oh(f) by A24,ASYMPT_0:16;
  then
A26: not f in Big_Omega(g) by ASYMPT_0:19;
  take f;
  f in Big_Oh(g) by A24,A25,ASYMPT_0:16;
  hence thesis by A26,Th4;
end;
