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 2
  not seq_n^(2) in Big_Omega(seq_n^(3))
proof
  set g = seq_n^(3);
  set f = seq_n^(2);
  now
    assume seq_n^(2) in Big_Omega(seq_n^(3));
    then consider s being Element of Funcs(NAT, REAL) such that
A1: s = f and
A2: ex d,N st d > 0 & for n st n >= N holds d*g.n <= s.n & s.n >= 0;
    consider d,N such that
A3: d > 0 and
A4: for n st n >= N holds d*g.n <= s.n & s.n >= 0 by A2;
A5: N+2 > 1+0 by XREAL_1:8;
    ex n st n >= N & d*g.n > s.n
    proof
      take n = max( N, [/(N+2)/d\] );
A6:   n >= N by XXREAL_0:25;
A7:   n is Integer by XXREAL_0:16;
A8:   [/(N+2)/d\] >= (N+2)/d by INT_1:def 7;
      (N+2)*d" > 0*d" by A3,XREAL_1:68;
      then
A9:   n > 0 by A8,XXREAL_0:25;
      reconsider n as Element of NAT by A6,A7,INT_1:3;
A10:  f.n*(n to_power -2) = (n to_power 2)*(n to_power -2) by A9,Def3
        .= (n to_power (2+(-2))) by A9,POWER:27
        .= 1 by POWER:24;
A11:  (n to_power -2) > 0 by A9,POWER:34;
A12:  d*n >= d*([/(N+2)/d\]) by A3,XREAL_1:64,XXREAL_0:25;
      d*([/(N+2)/d\]) >= d*((N+2)/d) by A3,A8,XREAL_1:64;
      then d*n >= ((N+2)/d)*d by A12,XXREAL_0:2;
      then
A13:  d*n >= N+2 by A3,XCMPLX_1:87;
      (d*g.n)*(n to_power -2) = d*(n to_power 3)*(n to_power -2) by A9,Def3
        .= d*((n to_power 3)*(n to_power -2))
        .= d*(n to_power (3+(-2))) by A9,POWER:27
        .= d*n by POWER:25;
      then (d*g.n)*(n to_power -2) > f.n*(n to_power -2) by A5,A10,A13,
XXREAL_0:2;
      hence thesis by A1,A11,XREAL_1:64,XXREAL_0:25;
    end;
    hence contradiction by A4;
  end;
  hence thesis;
end;
