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
  ex s being eventually-positive Real_Sequence st s = seq_a^(2,1,0) & 2
  (#)seq_n^(1) in Big_Oh( seq_n^(1) ) & not seq_a^(2,2,0) in Big_Oh(s)
proof
  reconsider q = seq_a^(2,1,0) as eventually-positive Real_Sequence;
  set p = seq_a^(2,2,0);
  set g = seq_n^(1);
  set f = 2(#)seq_n^(1);
  take q;
  thus q = seq_a^(2,1,0);
A1: now
    let n;
    assume n >= 0;
    thus f.n <= 2*g.n by SEQ_1:9;
A2: g.n = n
    proof
      per cases;
      suppose
        n = 0;
        hence thesis by Def3;
      end;
      suppose
        n > 0;
        hence g.n = n to_power 1 by Def3
          .= n by POWER:25;
      end;
    end;
    2*n >= 2*0;
    hence f.n >= 0 by A2,SEQ_1:9;
  end;
  f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
  hence f in Big_Oh(g) by A1;
  now
    assume p in Big_Oh(q);
    then consider t being Element of Funcs(NAT, REAL) such that
A3: t = p and
A4: ex c,N st c > 0 & for n st n >= N holds t.n <= c*q.n & t.n >= 0;
    consider c,N such that
A5: c > 0 and
A6: for n st n >= N holds t.n <= c*q.n & t.n >= 0 by A4;
    ex n st n >= N & t.n > c*q.n
    proof
      take n = max( N, [/log(2,c)+1\] );
A7:   n >= N by XXREAL_0:25;
      n is Integer by XXREAL_0:16;
      then reconsider n as Element of NAT by A7,INT_1:3;
A8:   2 to_power n >= 2 to_power [/log(2,c)+1\] by PRE_FF:8,XXREAL_0:25;
A9:   2 to_power -n > 0 by POWER:34;
      [/log(2,c)+1\] >= log(2,c)+1 by INT_1:def 7;
      then
A10:  2 to_power [/log(2,c)+1\] >= 2 to_power (log(2,c)+1) by PRE_FF:8;
A11:  2 to_power (log(2,c)+1) = (2 to_power log(2,c))*(2 to_power 1) by
POWER:27
        .= c*(2 to_power 1) by A5,POWER:def 3
        .= c*2 by POWER:25;
      (c*q.n)*(2 to_power -n) = (c*(2 to_power (1*n+0)))*(2 to_power -n)
      by Def1
        .= c*((2 to_power n)*(2 to_power -n))
        .= c*(2 to_power (n+-n)) by POWER:27
        .= c*1 by POWER:24;
      then 2 to_power (log(2,c)+1) > (c*q.n)*(2 to_power -n) by A5,A11,
XREAL_1:68;
      then
A12:  2 to_power [/log(2,c)+1\] > (c*q.n)*(2 to_power -n) by A10,XXREAL_0:2;
      p.n*(2 to_power -n) = (2 to_power (2*n+0))*(2 to_power -n) by Def1
        .= (2 to_power (2*n+(-1)*n)) by POWER:27
        .= (2 to_power (1*n));
      then p.n*(2 to_power -n) > (c*q.n)*(2 to_power -n) by A8,A12,XXREAL_0:2;
      hence thesis by A3,A9,XREAL_1:64,XXREAL_0:25;
    end;
    hence contradiction by A6;
  end;
  hence thesis;
end;
