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
  seq_n^(2) in Big_Oh(seq_n^(3))
proof
  set g = seq_n^(3);
  set f = seq_n^(2);
A1: now
    let n;
    assume
A2: n >= 2;
    then
A3: n > 1 by XXREAL_0:2;
A4: f.n = n to_power 2 by A2,Def3;
    g.n = n to_power 3 by A2,Def3;
    hence f.n <= 1*g.n by A3,A4,POWER:39;
    thus f.n >= 0 by A4;
  end;
  f is Element of Funcs(NAT, REAL) by FUNCT_2:8;
  hence thesis by A1;
end;
