
theorem LRNG1:
  for c be non empty positive-yielding XFinSequence of REAL
  ex a be Real, k,N be Nat
  st 0 < a & 0 < k
  & for x be Nat st N<= x holds (polynom(c)).x <=a* (x to_power k)
  proof
    let c be non empty positive-yielding XFinSequence of REAL;
    1 - 1 <= (len c) -1 by XREAL_1:9,NAT_1:14;then
    (len c) -1 in NAT by INT_1:3;then
    reconsider k = (len c) -1 as Nat;
    C1:len c = k+1;
    k+1 -1 < len c - 0 by XREAL_1:15;
    then
    k in Segm(len c) by NAT_1:44; then
    C2: c.k in rng c by FUNCT_1:3;
    for n being Nat st 0 <= n holds
    0 <= (seq_p(c)).n by NLM3;
    then
    reconsider f = seq_p(c) as
    eventually-nonnegative Real_Sequence by ASYMPT_0:def 2;
    f in Big_Oh( seq_n^(k) ) by ASYMPT_2:45,C1,C2,PARTFUN3:def 1;
    then
    consider N be Nat such that C5:
    for x be Nat st N <= x holds
    f.x <= (seq_n^(k+1)).x by ASYMPT_2:39;
    reconsider m = k+1 as Nat;
    reconsider M = N+1 as Nat;
    take 1,m,M;
    for x be Nat st M <= x holds f.x <= 1* (x to_power m)
    proof
      let x be Nat;
      assume C8:M <= x;
      CX: x is Element of NAT by ORDINAL1:def 12;
      N+1 -1 < M -0 by XREAL_1:15; then
      N < x by C8,XXREAL_0:2; then
      f.x <= (seq_n^(m)).x by C5;
      hence thesis by C8,ASYMPT_1:def 3,CX;
    end;
    hence thesis;
  end;
