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 holds f.n = Step1(n))
  ex s being eventually-positive Real_Sequence st s = f
  & s is not smooth & (for n holds f.n <= (seq_n^(1)).n)
  & f is eventually-nondecreasing
proof
  set g = seq_n^(1);
  let f be Real_Sequence such that
A1: for n holds f.n = Step1(n);
  f is eventually-positive
  proof
    take 1;
    let n be Nat;
A2:  n in NAT by ORDINAL1:def 12;
    assume n >= 1;
    then
A3: ex m st m! <= n & n < (m+1)! & Step1(n) = m! by Def6;
    f.n = Step1(n) by A1,A2;
    hence thesis by A3,NEWTON:17;
  end;
  then reconsider s=f as eventually-positive Real_Sequence;
  take s;
  thus s=f;
  now
    let k;
    thus f.k <= f.(k+1)
    proof
      per cases;
      suppose
A4:     k = 0;
A5:     f.(0+1) = Step1(1) by A1;
        f.0 = Step1(0) by A1
          .= 0 by Def6;
        hence thesis by A4,A5;
      end;
      suppose
        k > 0;
        then consider n1 such that
A6:     n1! <= k and
A7:     k < (n1+1)! and
A8:     Step1(k) = n1! by Def6;
A9:     k+1 <= (n1+1)! by A7,INT_1:7;
A10:     k <= k+1 by NAT_1:11;
A11:    f.k = n1! by A1,A8;
        per cases by A9,XXREAL_0:1;
        suppose
A12:      k+1 < (n1+1)!;
          n1! <= k+1 by A10,A6,XXREAL_0:2;
          then Step1(k+1) = n1! by A12,Def6;
          hence thesis by A1,A11;
        end;
        suppose
A13:      k+1 = (n1+1)!;
A14:      (n1+1)! > 0 by NEWTON:17;
          n1 + 2 > 0 + 1 by XREAL_1:8;
          then 1*((n1+1)!) < (n1+2)*((n1+1)!) by A14,XREAL_1:68;
          then
A15:      k+1 < ((n1+1)+1)! by A13,NEWTON:15;
          f.(k+1) = Step1(k+1) by A1
            .= (n1+1)! by A13,A15,Def6;
          hence thesis by A11,Lm51,NAT_1:11;
        end;
      end;
    end;
  end;
  then
A16: for k st k >= 0 holds f.k <= f.(k+1);
A17: 1 = 2-1;
  hereby
    set h = (f taken_every 2);
    assume s is smooth;
    then s is_smooth_wrt 2;
    then consider t being Element of Funcs(NAT, REAL) such that
A18: t = h and
A19: 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
    c > 0 and
A20: for n st n >= N holds t.n <= c*f.n & t.n >= 0 by A19;
    set n2 = max(max(N,3), [/c\]+1);
A21: n2 >= max(N,3) by XXREAL_0:25;
    max(N,3) >= N by XXREAL_0:25;
    then
A22: n2 >= N by A21,XXREAL_0:2;
A23: n2 >= [/c\]+1 by XXREAL_0:25;
A24: n2 is Integer by XXREAL_0:16;
A25: max(N,3) >= 3 by XXREAL_0:25;
    then
A26: n2 >= 3 by A21,XXREAL_0:2;
    reconsider n2 as Element of NAT by A21,A24,INT_1:3;
    set n1 = n2!-1;
A27: n2 > 2 by A26,XXREAL_0:2;
    then
A28: n2! >= 2 by Lm51,NEWTON:14;
    then
A29: n1 >= 1 by A17,XREAL_1:9;
    set n3 = n2-1;
    1+1 <= n2 by A26,XXREAL_0:2;
    then
A30: 1 <= n2-1 by XREAL_1:19;
A31: n3 >= 1 by A17,A27,XREAL_1:9;
    reconsider n1 as Element of NAT by A29,INT_1:3;
A32: t.n1 = f.(2*n1) by A18,ASYMPT_0:def 15;
    n2! > n2 by A26,Lm53;
    then n2! >= n2 + 1 by INT_1:7;
    then n1 >= n2 by XREAL_1:19;
    then n1 >= N by A22,XXREAL_0:2;
    then
A33: t.n1 <= c*f.n1 by A20;
    n2 < n2+1 by NAT_1:13;
    then n2*(n2!) < (n2+1)*(n2!) by A28,XREAL_1:68;
    then
A34: n2*(n2!) < (n2+1)! by NEWTON:15;
    n2! + 2 <= n2! + n2! by A28,XREAL_1:6;
    then
A35: n2! <= 2*(n2!) - 2*1 by XREAL_1:19;
A36: n2!-1 < n2!-0 by XREAL_1:15;
    then
A37: 2*n1 < 2*(n2!) by XREAL_1:68;
    reconsider n3 as Element of NAT by A31,INT_1:3;
    n3! >= 1 by A31,Lm51,NEWTON:13;
    then 1*1 <= (n2-1)*(n3!) by A30,Lm20;
    then n2*1 <= (n2-1)*(n3!)*n2 by XREAL_1:64;
    then n2 <= (n2-1)*((n3!)*(n3+1));
    then n2 <= (n2-1)*(n2!) by NEWTON:15;
    then
A38: n2! + n2 <= (n2!)*1 + (n2-1)*(n2!) by XREAL_1:6;
A39: n3+1 = n2+0;
    then n2*(n3!) = n2! by NEWTON:15;
    then n2*(n3!) <= n2*(n2!) - n2 by A38,XREAL_1:19;
    then n3! <= (n2*(n2!-1))/(n2*1) by A21,A25,XREAL_1:77;
    then
A40: n3! <= (n2! - 1)/1 by A21,A25,XCMPLX_1:91;
A41: [/c\] >= c by INT_1:def 7;
    [/c\] + 1 > [/c\] + 0 by XREAL_1:8;
    then [/c\] + 1 > c by A41,XXREAL_0:2;
    then
A42: n2 > c by A23,XXREAL_0:2;
A43: n3! > 0 by NEWTON:17;
    2*(n2!) <= n2*(n2!) by A27,XREAL_1:64;
    then 2*(n2!) < (n2+1)! by A34,XXREAL_0:2;
    then
A44: 2*n1 < (n2+1)! by A37,XXREAL_0:2;
A45: f.(2*n1) = Step1(2*n1) by A1
      .= (n3+1)! by A28,A44,A35,Def6
      .= n2*(n3!) by NEWTON:15;
    f.n1 = Step1(n1) by A1
      .= n3! by A29,A36,A39,A40,Def6;
    hence contradiction by A33,A32,A45,A42,A43,XREAL_1:68;
  end;
  hereby
    let n;
    thus f.n <= g.n
    proof
      per cases;
      suppose
A46:    n = 0;
        f.0 = Step1(0) by A1
          .= 0 by Def6;
        hence thesis by A46,Def3;
      end;
      suppose
A47:    n > 0;
        then
A48:    g.n = n to_power 1 by Def3
          .= n by POWER:25;
        ex n1 st n1! <= n & n < (n1+1)! & Step1(n) = n1! by A47,Def6;
        hence thesis by A1,A48;
      end;
    end;
  end;
  reconsider zz=0 as Nat;
  take zz;
  let n be Nat;
   n in NAT by ORDINAL1:def 12;
  hence thesis by A16;
end;
