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 > 0 holds f.n = (n to_power
sqrt n)) holds ex s,s1 being eventually-positive Real_Sequence st s = f & s1 =
  seq_a^(2,1,0) & Big_Oh(s) c= Big_Oh(s1) & not Big_Oh(s) = Big_Oh(s1)
proof
  set g = seq_a^(2,1,0);
  let f be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power sqrt n);
A2: f is eventually-positive
  proof
    take 1;
    let n be Nat;
A3:  n in NAT by ORDINAL1:def 12;
    assume
A4: n >= 1;
    then f.n = n to_power sqrt n by A1,A3;
    hence thesis by A4,POWER:34;
  end;
  set h = f/"g;
  reconsider f as eventually-positive Real_Sequence by A2;
  reconsider g as eventually-positive Real_Sequence;
  take f,g;
  consider N such that
A5: for n st n >= N holds n - sqrt n*log(2,n) > n/2 by Lm40;
A6: now
    let p be Real;
    assume
A7: p > 0;
    set N1 = max( N, max(2*[/log(2,1/p)\], 2) );
A8: N1 >= N by XXREAL_0:25;
A9: N1 is Integer
    proof
      per cases by XXREAL_0:16;
      suppose
        N1 = N;
        hence thesis;
      end;
      suppose
        N1 = max(2*[/log(2,1/p)\], 2);
        hence thesis by XXREAL_0:16;
      end;
    end;
A10: N1 >= max(2*[/log(2,1/p)\], 2) by XXREAL_0:25;
    max(2*[/log(2,1/p)\], 2) >= 2*[/log(2,1/p)\] by XXREAL_0:25;
    then
A11: N1 >= 2*[/log(2,1/p)\] by A10,XXREAL_0:2;
    N1 in NAT by A8,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;
A14: [/log(2,1/p)\] >= log(2,1/p) by INT_1:def 7;
    assume
A15: n >= N1;
    then n >= 2*[/log(2,1/p)\] by A11,XXREAL_0:2;
    then n/2 >= [/log(2,1/p)\] by XREAL_1:77;
    then n/2 >= log(2,1/p) by A14,XXREAL_0:2;
    then -(n/2) <= -log(2,1/p) by XREAL_1:24;
    then 2 to_power -(n/2) <= 2 to_power -log(2,1/p) by PRE_FF:8;
    then 2 to_power -(n/2) <= 1/(2 to_power log(2,1/p)) by POWER:28;
    then
A16: 2 to_power -(n/2) <= 1/(1/p) by A7,POWER:def 3;
A17: g.n = 2 to_power (1*n+0) by Def1
      .= 2 to_power n;
A18: max(2*[/log(2,1/p)\], 2) >= 2 by XXREAL_0:25;
    then f.n = n to_power sqrt n by A1,A10,A15,A12
      .= 2 to_power (sqrt n*log(2,n)) by A10,A18,A15,Lm3;
    then
A19: h.n = 2 to_power ((sqrt n*log(2,n)) - n) by A13,A17,POWER:29
      .= 2 to_power -(n - sqrt n*log(2,n));
    then
A20: h.n > 0 by POWER:34;
    n >= N by A8,A15,XXREAL_0:2;
    then n - sqrt n*log(2,n) > n/2 by A5,A12;
    then -(n - sqrt n*log(2,n)) < -(n/2) by XREAL_1:24;
    then 2 to_power -(n - sqrt n*log(2,n)) < 2 to_power -(n/2) by POWER:39;
    then h.n < p by A19,A16,XXREAL_0:2;
    hence |.h.n-0.| < p by A20,ABSVALUE:def 1;
  end;
  then
A21: h is convergent by SEQ_2:def 6;
  then
A22: lim h = 0 by A6,SEQ_2:def 7;
  then not g in Big_Oh(f) by A21,ASYMPT_0:16;
  then
A23: not f in Big_Omega(g) by ASYMPT_0:19;
  f in Big_Oh(g) by A21,A22,ASYMPT_0:16;
  hence thesis by A23,Th4;
end;
