
theorem LRNG2:
  for a be Real, k be Nat holds
  ex c be non empty positive-yielding XFinSequence of REAL
  st for x be Nat holds a* (x to_power k) <= (polynom(c)).x
  proof
    let a be Real,k be Nat;
    reconsider c = Segm (k+1) --> (|. a .| +1 ) as XFinSequence of REAL;
    reconsider c as non empty positive-yielding XFinSequence of REAL;
    take c;
    for x be Nat holds a* (x to_power k) <=(polynom(c)).x
    proof
      let x be Nat;
      set c2 = c (#) seq_a^(x,1,0);
      T0: (polynom(c)).x = Sum(c2) by ASYMPT_2:def 2;
      LN2:k +0 < k+1 by XREAL_1:8;
      T1:dom c2 = dom c /\ dom seq_a^(x,1,0) by VALUED_1:def 4
      .= dom c /\ NAT by SEQ_1:1
      .= (Segm (k+1)) /\ NAT;
      T3:c2.k=(c.k)*((seq_a^(x,1,0)).k) by VALUED_1:5
      .=(c.k)*(x to_power (1*k+0)) by ASYMPT_1:def 1
      .= (|. a .| +1)*(x to_power k) by FUNCOP_1:7,NAT_1:44,LN2;
      a < (|. a .| +1) by TCL001; then
      T4:a*(x to_power k) <= (|. a .| +1)*(x to_power k) by XREAL_1:64;
      len c2 = k+1 by T1,XBOOLE_1:28;then
      Sum c2 >= (|. a .| +1)*(x to_power k) by T3,AFINSQ_2:61,NAT_1:44,LN2;
      hence thesis by T4,XXREAL_0:2,T0;
    end;
    hence thesis;
  end;
