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 f being Real_Sequence st (for n st n > 0 holds f.n = (n to_power
log(2,n))) holds ex s being eventually-positive Real_Sequence st s = f & not s
  is smooth
proof
  let f be Real_Sequence such that
A1: for n st n > 0 holds f.n = (n to_power log(2,n));
A2: f is eventually-positive
  proof
    take 1;
    let n be Nat;
A3:  n in NAT by ORDINAL1:def 12;
    assume
A4: n >= 1;
    then f.n = n to_power log(2,n) by A1,A3;
    hence thesis by A4,POWER:34;
  end;
  set g = f taken_every 2;
  reconsider f as eventually-positive Real_Sequence by A2;
  take f;
  now
    assume f is smooth;
    then f is_smooth_wrt 2;
    then consider t being Element of Funcs(NAT, REAL) such that
A5: t = g and
A6: ex c,N st c > 0 & for n st n >= N holds t.n <= c*f.n & t.n >= 0;
    consider c,N such that
A7: c > 0 and
A8: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A6;
A9: sqrt c > 0 by A7,SQUARE_1:25;
    set N0 = [/sqrt c / sqrt 2\];
    reconsider N2=max(N, N0) as Integer by XXREAL_0:16;
    set N1 = max( N2, 2 );
A10: N1 >= N2 by XXREAL_0:25;
    N2 >= N0 by XXREAL_0:25;
    then
A11: N1 >= N0 by A10,XXREAL_0:2;
A12: N1 is Integer by XXREAL_0:16;
    N2 >= N by XXREAL_0:25;
    then
A13: N1 >= N by A10,XXREAL_0:2;
    N1 >= 2 by XXREAL_0:25;
    then reconsider N1 as Element of NAT by A12,INT_1:3;
    set n = N1 + 1;
A14: n to_power log(2,n) > 0 by POWER:34;
A15: 2*n > 2*0 by XREAL_1:68;
A16: sqrt 2 <> 0 by SQUARE_1:25;
A17: sqrt 2 > 0 by SQUARE_1:25;
A18: N0 >= sqrt c / sqrt 2 by INT_1:def 7;
A19: n > N1 + 0 by XREAL_1:8;
    then n > N0 by A11,XXREAL_0:2;
    then n > sqrt c / sqrt 2 by A18,XXREAL_0:2;
    then n*sqrt 2 > (sqrt c / sqrt 2) * sqrt 2 by A17,XREAL_1:68;
    then n*sqrt 2 > sqrt c by A16,XCMPLX_1:87;
    then (n*sqrt 2)^2 > (sqrt c)^2 by A9,SQUARE_1:16;
    then n^2*(sqrt 2)^2 > c by A7,SQUARE_1:def 2;
    then
A20: 2*n^2 > c by SQUARE_1:def 2;
    (2*n^2)*(n to_power log(2,n)) = ((2*n)*n)*(n to_power log(2,n))
      .= ((2*n)*(2 to_power (log(2,n))))*(n to_power log(2,n)) by POWER:def 3
      .= (2*n)*((2 to_power (log(2,n)))*(n to_power log(2,n)))
      .= (2*n)*((2*n) to_power log(2,n)) by POWER:30
      .= ((2*n) to_power 1)*((2*n) to_power log(2,n)) by POWER:25
      .= (2*n) to_power (1 + log(2,n)) by A15,POWER:27
      .= (2*n) to_power (log(2,2) + log(2,n)) by POWER:52
      .= (2*n) to_power log(2,2*n) by POWER:53;
    then (2*n) to_power log(2,2*n) > c*(n to_power log(2,n)) by A14,A20,
XREAL_1:68;
    then f.(2*n) > c*(n to_power log(2,n)) by A1,A15;
    then t.n > c*(n to_power log(2,n)) by A5,ASYMPT_0:def 15;
    then
A21: t.n > c*f.n by A1;
    n > N by A13,A19,XXREAL_0:2;
    hence contradiction by A8,A21;
  end;
  hence thesis;
end;
