
theorem RNG2:
  for g be polynomially-abs-bounded Function of NAT,REAL
  ex d be non empty positive-yielding XFinSequence of REAL,
  N be Nat st for x be Nat
  st N<= x holds |. g.x .| <= (polynom(d)).x
  proof
    let g be polynomially-abs-bounded Function of NAT,REAL;
    consider k1 be Nat such that
    A4: |.g.| in Big_Oh(seq_n^(k1)) by defabs;
    consider t1 be Element of Funcs(NAT, REAL) such that
    A6: |.g.|=t1 &
    ex c1 be Real,N1 be Element of NAT
    st c1 > 0
    & for n be Element of NAT
    st n >= N1 holds t1.n <= c1*(seq_n^(k1)).n
    & t1.n >= 0 by A4;
    consider c1 be Real,N1 be Element of NAT such that
    A7: c1 > 0
    & for n be Element of NAT
    st n >= N1 holds t1.n <= c1*(seq_n^(k1)).n
    & t1.n >= 0 by A6;
    consider d be non empty positive-yielding
    XFinSequence of REAL such that
    A8: for x be Nat holds
    c1* (x to_power k1) <=(polynom(d)).x by LRNG2;
    reconsider N=N1+1 as Nat;
    take d,N;
    let x be Nat;
    assume
    B1: N<=x;
    LXN: x is Element of NAT by ORDINAL1:def 12;
    N1<=N by NAT_1:12; then
    N1<=x by XXREAL_0:2,B1; then
    A9:t1.x <= c1*(seq_n^(k1)).x & t1.x >= 0 by A7,LXN;
    A10:(seq_n^(k1)).x = x to_power k1
    by ASYMPT_1:def 3,LXN,B1;
    c1* (x to_power k1) <=(polynom(d)).x by A8; then
    t1.x <= (polynom(d)).x by A10,A9,XXREAL_0:2;
    hence |.g.x.| <= (polynom(d)).x by A6,VALUED_1:18;
  end;
