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 4, O(2^n) c O(2^2n))
  ex s, s1 being eventually-positive Real_Sequence st s = seq_a^(2,1,0)
  & s1 = seq_a^(2,2,0) & Big_Oh(s) c= Big_Oh(s1) & not Big_Oh(s) = Big_Oh(s1)
proof
  reconsider g = seq_a^(2,2,0) as eventually-positive Real_Sequence;
  reconsider f = seq_a^(2,1,0) as eventually-positive Real_Sequence;
  take f,g;
  thus f = seq_a^(2,1,0) & g = seq_a^(2,2,0);
  set h = f/"g;
A1: for n holds h.n = (2 to_power -n)
  proof
    let n;
    h.n = f.n/g.n by Lm4
      .= (2 to_power (1*n+0))/g.n by Def1
      .= (2 to_power (1*n))/(2 to_power (2*n+0)) by Def1
      .= 2 to_power (1*n-(2*n)) by POWER:29
      .= 2 to_power -n;
    hence thesis;
  end;
A2: now
    let p be Real;
    set N = max(1,[/log(2,1/p)\] + 1);
A3: N >= 1 by XXREAL_0:25;
A4: N is Integer by XXREAL_0:16;
A5: [/log(2,1/p)\] >= log(2,1/p) by INT_1:def 7;
    [/log(2,1/p)\] + 1 > [/log(2,1/p)\] by XREAL_1:29;
    then [/log(2,1/p)\] + 1 > log(2,1/p) by A5,XXREAL_0:2;
    then
A6: 2 to_power ([/log(2,1/p)\]+1) > 2 to_power log(2,1/p) by POWER:39;
    N in NAT by A3,A4,INT_1:3;
    then reconsider N as Nat;
    assume
A7: p > 0;
    take N;
    let n be Nat;
A8: n in NAT by ORDINAL1:def 12;
    [/log(2,1/p)\] + 1 <= N by XXREAL_0:25; then
    2 to_power N >= 2 to_power ([/log(2,1/p)\] + 1) by PRE_FF:8;
    then
A9: 2 to_power N > 2 to_power log(2,1/p) by A6,XXREAL_0:2;
    assume n >= N;
    then 2 to_power n >= 2 to_power N by PRE_FF:8;
    then 2 to_power n > 2 to_power log(2,1/p) by A9,XXREAL_0:2;
    then 2 to_power n > 1/p by A7,POWER:def 3;
    then (2 to_power n)*p > 1/p*p by A7,XREAL_1:68;
    then
A10: p*(2 to_power n) > 1 by A7,XCMPLX_1:87;
    2 to_power n > 0 by POWER:34;
    then (p*(2 to_power n))*(2 to_power n)" > 1*(2 to_power n)" by A10,
XREAL_1:68;
    then
A11: p*((2 to_power n)*(2 to_power n)") > (2 to_power n)";
    2 to_power n <> 0 by POWER:34;
    then p*1 > (2 to_power n)" by A11,XCMPLX_0:def 7;
    then
A12: p > 1/(2 to_power n);
A13: (2 to_power -n) > 0 by POWER:34;
    |.h.n-0.| = |.(2 to_power -n).| by A1,A8;
    then |.h.n-0.| = (2 to_power -n) by A13,ABSVALUE:def 1;
    hence |.h.n-0.| < p by A12,POWER:28;
  end;
  then
A14: h is convergent by SEQ_2:def 6;
  then
A15: lim h = 0 by A2,SEQ_2:def 7;
  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 A14,A15,ASYMPT_0:16;
  hence thesis by A16,Th4;
end;
