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(n^logn) c O(n^sqrt n))
  for f,g being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
  log(2,n))) & (for n st n > 0 holds g.n = (n to_power sqrt n)) holds ex s,s1
being eventually-positive Real_Sequence st s = f & s1 = g & Big_Oh(s) c= Big_Oh
  (s1) & not Big_Oh(s) = Big_Oh(s1)
proof
  let f,g be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power log(2,n)) and
A2: for n st n > 0 holds g.n = (n to_power sqrt n);
  set h = f/"g;
  g is eventually-positive
  proof
    take 1;
    let n be Nat;
A3:  n in NAT by ORDINAL1:def 12;
    assume
A4: n >= 1;
    then g.n = n to_power sqrt n by A2,A3;
    hence thesis by A4,POWER:34;
  end;
  then reconsider g as eventually-positive Real_Sequence;
  f is eventually-positive
  proof
    take 1;
    let n be Nat;
A5:  n in NAT by ORDINAL1:def 12;
    assume
A6: n >= 1;
    then f.n = n to_power log(2,n) by A1,A5;
    hence thesis by A6,POWER:34;
  end;
  then reconsider f as eventually-positive Real_Sequence;
  take f,g;
  consider N such that
A7: for n st n >= N holds sqrt n - log(2, n) > 1 by Lm32;
A8: now
    let p be Real such that
A9: p > 0;
    set N1 = max( N, max([/1/p\], 2) );
A10: N1 >= N by XXREAL_0:25;
A11: N1 is Integer
    proof
      per cases by XXREAL_0:16;
      suppose
        N1 = N;
        hence thesis;
      end;
      suppose
        N1 = max([/1/p\], 2);
        hence thesis by XXREAL_0:16;
      end;
    end;
A12: N1 >= max([/1/p\], 2) by XXREAL_0:25;
    max([/1/p\], 2) >= [/1/p\] by XXREAL_0:25;
    then
A13: N1 >= [/1/p\] by A12,XXREAL_0:2;
A14: max([/1/p\], 2) >= 2 by XXREAL_0:25;
    then N1 >= 2 by A12,XXREAL_0:2;
    then
A15: N1 > 1 by XXREAL_0:2;
    N1 in NAT by A10,A11,INT_1:3;
    then reconsider N1 as Nat;
    take N1;
    let n be Nat;
A16:  n in NAT by ORDINAL1:def 12;
A17: h.n = f.n/g.n by Lm4;
    assume
A18: n >= N1;
    then f.n = n to_power log(2,n) by A1,A12,A14,A16;
    then
A19: h.n = (n to_power log(2,n)) / (n to_power sqrt n)
          by A2,A12,A14,A18,A17,A16
      .= n to_power (log(2,n) - sqrt n) by A12,A14,A18,POWER:29
      .= n to_power -(sqrt n - log(2,n));
    then
A20: h.n > 0 by A12,A14,A18,POWER:34;
    n >= N by A10,A18,XXREAL_0:2;
    then sqrt n - log(2, n) > 1 by A7,A16;
    then
A21: (-1)*(sqrt n - log(2, n)) < (-1)*1 by XREAL_1:69;
    n > 1 by A15,A18,XXREAL_0:2;
    then
A22: n to_power -(sqrt n - log(2,n)) < n to_power -1 by A21,POWER:39;
    [/1/p\] >= 1/p by INT_1:def 7;
    then N1 >= 1/p by A13,XXREAL_0:2;
    then n >= 1/p by A18,XXREAL_0:2;
    then
A23: 1/n <= 1/(1/p) by A9,XREAL_1:85;
    n to_power -1 = 1/(n to_power 1) by A12,A14,A18,POWER:28
      .= 1/n by POWER:25;
    then h.n < p by A19,A22,A23,XXREAL_0:2;
    hence |.h.n-0.| < p by A20,ABSVALUE:def 1;
  end;
  then
A24: h is convergent by SEQ_2:def 6;
  then
A25: lim h = 0 by A8,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;
  f in Big_Oh(g) by A24,A25,ASYMPT_0:16;
  hence thesis by A26,Th4;
end;
