
theorem LMXFIN10:
  for k be Nat,
      c be nonnegative-yielding XFinSequence of REAL
  st len c = k+1
  holds seq_p(c) in Big_Oh( seq_n^(k) )
  proof
    defpred P[Nat] means
    for c be nonnegative-yielding XFinSequence of REAL
    st len c = $1+1
    holds seq_p(c) in Big_Oh( seq_n^($1) );
    P0:P[0] by LMXFIN6;
    P1:for k be Nat st P[k] holds P[k+1]
    proof
      let k be Nat;
      assume P11: P[k];
      let d be nonnegative-yielding XFinSequence of REAL;
      assume P12: len d = (k+1)+1;then
      consider a be Real,d1 be XFinSequence of REAL,
      y be Real_Sequence such that
      P13: len d1 = k + 1 & d1= d | (k+1) &
      a = d.(k+1) & d =d1^<% a %> & seq_p(d) = seq_p(d1) + y &
      for i be Nat holds
      y.i = a* (i to_power (k+1)) by LMXFIN4;
      T11: for i be Nat st i in dom d1 holds 0 <=d1.i
      proof
        let i be Nat;
        assume AA1: i in dom d1; then
        AA2: d1.i = d.i by P13,FUNCT_1:47;
        k+1 <= (k+1)+1 by NAT_1:13; then
        Segm (k+1) c= Segm ((k+1)+1) by NAT_1:39;
        hence 0 <=d1.i by PARTFUN3:def 4,AA2,FUNCT_1:3,AA1,P12,P13;
      end;
      for r be Real st r in rng d1 holds 0 <=r
      proof
        let r be Real;
        assume r in rng d1;then
        consider x be object such that RC:
        x in dom d1 & r = d1.x by FUNCT_1:def 3;
        thus thesis by RC,T11;
      end; then
      d1 is nonnegative-yielding;then
      seq_p(d1) in Big_Oh( seq_n^k ) by P11,P13; then
      P14: seq_p(d1) in Big_Oh( seq_n^(k+1)) by LMXFIN9,TARSKI:def 3;
      k+1 < (k+1)+1 by NAT_1:13; then
      k+1 in Segm ((k+1)+1) by NAT_1:44;then
      d.(k+1) in rng d by FUNCT_1:3,P12;then
      y in Big_Oh( seq_n^(k+1) ) by P13,LMXFIN8,PARTFUN3:def 4;
      hence seq_p(d) in Big_Oh( seq_n^(k+1) ) by P14,P13,LMXFIN7;
    end;
    for k be Nat holds P[k] from NAT_1:sch 2(P0,P1);
    hence thesis;
  end;
