
theorem LMXFIN20A:
  for k be Nat,
  c be XFinSequence of REAL
  st len c = k+1 & seq_p(c) is eventually-nonnegative
  holds seq_p(c) in Big_Oh( seq_n^(k) )
  proof
    let k be Nat,
    c be XFinSequence of REAL;
    assume AS: len c = k+1 & seq_p(c) is eventually-nonnegative;
    consider d be XFinSequence of REAL such that
    P1: len d = len c &
    for i be Nat st i in dom d holds d.i = |. c.i .| by LMXFIN15;
    T11: for i be Nat st i in dom d holds 0 <= d.i
    proof
      let i be Nat;
      assume i in dom d;
      then d.i = |. c.i .| by P1;
      hence 0 <= d.i by COMPLEX1:46;
    end;
    for r be Real st r in rng d holds 0 <=r
    proof
      let r be Real;
      assume r in rng d;then
      consider x be object such that RC:
      x in dom d & r = d.x by FUNCT_1:def 3;
      thus thesis by RC,T11;
    end;
    then
    d is nonnegative-yielding;
    then
    seq_p(d) in Big_Oh( seq_n^(k) ) by LMXFIN10,P1,AS; then
    consider t be Element of Funcs (NAT,REAL) such that
    P5: seq_p(d)=t &
    ex c being Real,N being Element of NAT st
    c > 0 & for n being Element of NAT st n >= N holds
    ( t . n <= c * ((seq_n^k).n) & t . n >= 0 );
    consider N1 be Nat such that
    P4A: for n be Nat st N1 <= n holds
    0 <= ( seq_p(c) ).n by AS;
    consider a being Real,N2 being Element of NAT such that
    P6: a > 0 &
    for n being Element of NAT st n >= N2 holds
    ( t . n <= a * ((seq_n^k).n) & (t . n >= 0 )) by P5;
    set N = N1+N2;
    P7: for n being Element of NAT st n >= N holds
    ((seq_p(c)).n <= a * ((seq_n^k).n) & (seq_p(c)).n >= 0)
    proof
      let n be Element of NAT;
      assume P8: n >= N;
      N1 <= N1 + N2 & N2 <= N1 + N2 by NAT_1:11; then
      P9: N1 <= n & N2 <= n by P8,XXREAL_0:2; then
      P10: ( seq_p(c) ).n <= ( seq_p(d) ).n &
      0 <= ( seq_p(c) ).n by P4A,P1,LMXFIN17;
      (seq_p(d)) . n <= a * ((seq_n^k).n)
      & ((seq_p(d)) . n >= 0 ) by P5,P9,P6;
      hence (seq_p(c)) . n <= a * ((seq_n^k).n) by P10,XXREAL_0:2;
      thus 0 <= ( seq_p(c) ).n by P4A,P9;
    end;
    seq_p(c) is Element of Funcs (NAT,REAL) by FUNCT_2:8;
    hence seq_p(c) in Big_Oh( seq_n^k ) by P6,P7;
  end;
