
theorem
  for a be Nat st 1 < a holds
    seq_a^(a,1,0) is non polynomially-bounded
  proof
    let a be Nat;
    assume AS1: 1 < a;
    assume seq_a^(a,1,0) is polynomially-bounded; then
    consider k be Nat such that CL1:
    seq_a^(a,1,0) in Big_Oh(seq_n^(k));
    reconsider f = seq_n^(k) as eventually-positive Real_Sequence;
    reconsider t = seq_a^(a,1,0)
    as eventually-nonnegative Real_Sequence by AS1;
    t in Big_Oh(f) &
    for n be Element of NAT st 1 <= n holds 0< (f).n by LC4,CL1;
    then consider c be Real such that
    LL1: c > 0 & for n be Element of NAT st n >= 1 holds
    seq_a^(a,1,0).n <= c*(seq_n^(k)).n by ASYMPT_0:8;
    TLCPP:for n be Nat st n >= 1 holds
    2 to_power n <= c*(n to_power k)
    proof
      let n be Nat;
      ZZ: n in NAT by ORDINAL1:def 12;
      assume AT1:n >= 1;then
      seq_a^(a,1,0).n <= c*((seq_n^(k)).n) by ZZ,LL1;then
      a to_power (1*n +0) <= c*((seq_n^(k)).n) by ASYMPT_1:def 1; then
      TZ1:a to_power (n) <= c*(n to_power k) by ASYMPT_1:def 3,AT1;
      1+1 <= a by AS1,INT_1:7;then
      2 to_power n <= a to_power n by LEMC01;
      hence thesis by XXREAL_0:2,TZ1;
    end;
    TLCPP2:ex N,b be Nat st
    for n be Nat st N <= n holds
    2 to_power n <= b*(n to_power k)
    proof
      consider N be Nat such that TLCPP3:
      for n be Nat st N <= n holds
      2 to_power n <= c*(n to_power k) by TLCPP;
      set b = [/ c \];
      TLCPP4: c <= b by INT_1:def 7;then
      reconsider b as Element of NAT by INT_1:3, LL1;
      take N,b;
      for n be Nat st N <= n holds
      2 to_power n <= b*(n to_power k)
      proof
        let n be Nat;
        assume N <= n;then
        TLCPP5: 2 to_power n <= c*(n to_power k) by TLCPP3;
        c*(n to_power k) <= b*(n to_power k) by XREAL_1:64,TLCPP4;
        hence thesis by TLCPP5, XXREAL_0:2;
      end;
      hence thesis;
    end;
    per cases;
    suppose 1 < k;
      hence contradiction by TLCPP2,N2POWINPOLY;
    end;
    suppose k <= 1;
      then TLCPPAA: k < 2 by XXREAL_0:2;
      ex N,b be Nat st
      for n be Nat st N <= n holds 2 to_power n <= b*(n to_power 2)
      proof
        consider N,b be Nat such that TLCPPA3:
        for n be Nat st N <= n holds
        2 to_power n <= b*(n to_power k) by TLCPP2;
        reconsider M = N+2 as Element of NAT;
        TLCPPAA1: N <= M by NAT_1:11;
        take M, b;
        let n be Nat;
        assume TLCPPAS: M <= n;then
        N <= n by XXREAL_0:2,TLCPPAA1;then
        TLCPPA4: 2 to_power n <= b*(n to_power k) by TLCPPA3;
        2 <= N +2 by NAT_1:11;then
        1 +1 <=n by TLCPPAS,XXREAL_0:2;then
        1 <n by NAT_1:13;then
        (n to_power k) <= (n to_power 2) by POWER:39,TLCPPAA;then
        b*(n to_power k) <= b*(n to_power 2) by XREAL_1:64;
        hence thesis by TLCPPA4, XXREAL_0:2;
      end;
      hence contradiction by N2POWINPOLY;
    end;
  end;
