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
  log(2,3) < 159/100 implies seq_n^(log(2,3)) in Big_Oh(seq_n^(159/100))
& not seq_n^(log(2,3)) in Big_Omega(seq_n^(159/100)) & not seq_n^(log(2,3)) in
  Big_Theta(seq_n^(159/100))
proof
  set c = 159/100 - log(2,3);
  set g = seq_n^(159/100);
  set f = seq_n^(log(2,3));
  set h = f/"g;
  assume
A1: log(2,3) < 159/100;
  then
A2: log(2,3) - log(2,3) < 159/100 - log(2,3) by XREAL_1:9;
A3: c/2 <> 0 by A1;
A4: now
A5: c*(1/2) < c*1 by A2,XREAL_1:68;
    let p be Real such that
A6: p > 0;
    reconsider p1 = p as Real;
A7: ((1/p1) to_power (1/(c/2))) > 0 by A6,POWER:34;
    set N1 = max([/((1/p1) to_power (1/(c/2)))\], 2);
A8: N1 >= [/((1/p) to_power (1/(c/2)))\] by XXREAL_0:25;
A9: N1 is Integer by XXREAL_0:16;
A10: N1 >= 2 by XXREAL_0:25;
    then
A11: N1 > 1 by XXREAL_0:2;
    N1 in NAT by A10,A9,INT_1:3;
    then reconsider N1 as Nat;
    take N1;
    let n be Nat;
A12:  n in NAT by ORDINAL1:def 12;
A13: h.n = f.n/g.n by Lm4;
    assume
A14: n >= N1;
    then f.n = (n to_power log(2,3)) by A10,Def3,A12;
    then
A15: h.n = (n to_power log(2,3)) / (n to_power (159/100))
               by A10,A14,A13,Def3,A12
      .= (n to_power (log(2,3) - (159/100))) by A10,A14,POWER:29
      .= (n to_power -c);
    [/((1/p) to_power (1/(c/2)))\]>= ((1/p) to_power (1/(c/2))) by INT_1:def 7;
    then N1 >= ((1/p) to_power (1/(c/2))) by A8,XXREAL_0:2;
    then n >= ((1/p) to_power (1/(c/2))) by A14,XXREAL_0:2;
    then n to_power (c/2) >= ((1/p) to_power (1/(c/2))) to_power (c/2) by A2,A7
,Lm6;
    then n to_power (c/2) >= (1/p1) to_power ((1/(c/2))*(c/2)) by A6,POWER:33;
    then n to_power (c/2) >= (1/p) to_power 1 by A3,XCMPLX_1:87;
    then n to_power (c/2) >= 1/p1 by POWER:25;
    then 1 / (n to_power (c/2)) <= 1 / (p") by A6,XREAL_1:85;
    then
A16: n to_power -(c/2) <= p by A10,A14,POWER:28;
    n > 1 by A11,A14,XXREAL_0:2;
    then
A17: n to_power (c/2) < n to_power c by A5,POWER:39;
    n to_power (c/2) > 0 by A10,A14,POWER:34;
    then 1 / (n to_power (c/2)) > 1 / (n to_power c) by A17,XREAL_1:88;
    then n to_power -(c/2) > 1 / (n to_power c) by A10,A14,POWER:28;
    then h.n < n to_power -(c/2) by A10,A14,A15,POWER:28;
    then
A18: h.n < p by A16,XXREAL_0:2;
    h.n > 0 by A10,A14,A15,POWER:34;
    hence |.h.n-0.| < p by A18,ABSVALUE:def 1;
  end;
  then
A19: h is convergent by SEQ_2:def 6;
  then
A20: lim h = 0 by A4,SEQ_2:def 7;
  hence f in Big_Oh(g) by A19,ASYMPT_0:16;
A21: not g in Big_Oh(f) by A19,A20,ASYMPT_0:16;
  hence not f in Big_Omega(g) by ASYMPT_0:19;
  not f in Big_Omega(g) by A21,ASYMPT_0:19;
  hence thesis by XBOOLE_0:def 4;
end;
