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 g being Real_Sequence st (for n holds g.n = (n^2 - n + 1) to_power
4) holds ex s being eventually-positive Real_Sequence st s = g & Big_Oh(seq_n^(
  8)) = Big_Oh(s)
proof
  let g be Real_Sequence such that
A1: for n holds g.n = (n^2 - n + 1) to_power 4;
  g is eventually-positive
  proof
    take 0;
    let n be Nat;
A2:  n in NAT by ORDINAL1:def 12;
    assume n >= 0;
    g.n = (n^2 - n + 1) to_power 4 by A1,A2;
    hence thesis by Lm21,POWER:34,A2;
  end;
  then reconsider g as eventually-positive Real_Sequence;
  take g;
  set f = seq_n^(8);
A3: now
    let n;
A4: g.n = (n^2 -n + 1) to_power 4 by A1;
    assume
A5: n >= 1;
    then
A6: n^2 -n + 1 <= n^2 by Lm23;
    f.n = n to_power (2*4) by A5,Def3
      .= n to_power 2 to_power 4 by A5,POWER:33
      .= n^2 to_power 4 by POWER:46;
    hence g.n <= 1*f.n by A4,A6,Lm6,Lm21;
    thus g.n >= 0 by A4,Lm21,POWER:34;
  end;
A7: now
    let n;
A8: g.n = (n^2 -n + 1) to_power 4 by A1;
A9: (n^2 -n + 1) > 0 by Lm21;
    assume
A10: n >= 1;
    then
A11: f.n = n to_power (2*4) by Def3
      .= n to_power 2 to_power 4 by A10,POWER:33
      .= n^2 to_power 4 by POWER:46;
A12: n*n > n*0 by A10,XREAL_1:68;
    n^2 <= 2*(n^2 -n + 1) by A10,Lm24;
    then f.n <= (2*(n^2 -n + 1)) to_power 4 by A11,A12,Lm6;
    hence f.n <= 16*g.n by A8,A9,POWER:30,62;
    thus f.n >= 0 by A11,A12,POWER:34;
  end;
  f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
  then
A13: f in Big_Oh(g) by A7;
  g is Element of Funcs(NAT, REAL) by FUNCT_2:8;
  then g in Big_Oh(f) by A3;
  hence thesis by A13,Lm5;
end;
