
theorem TLNEG3:
  for k be Nat, d be XFinSequence of REAL
  st len d = k+1 & 0 < d.k holds
  seq_p(d) is eventually-nonnegative
  proof
    let k be Nat, d be XFinSequence of REAL;
    assume
    AS: len d = k+1 & 0 < d.k; then
    consider a be Real,d1 be XFinSequence of REAL,
    y be Real_Sequence such that
    P1: len d1 = k & d1= d | k & a = d.k & d = d1^<% a %> &
    seq_p(d) = seq_p(d1) + y &
    for i be Nat holds y.i = a* (i to_power k) by LMXFIN4;
    consider N be Nat such that
    P20: for i be Nat st N <= i holds |. (seq_p(d1)).i .| <= y.i
    by P1,TLNEG35,AS;
    for i be Nat st N <= i holds 0 <= (seq_p(d)).i
    proof
      let i be Nat;
      assume N <= i; then
      P32: 0 <= y.i - |. (seq_p(d1)).i .| by XREAL_1:48,P20;
      -(seq_p(d1)).i <= -(-|. (seq_p(d1)).i .|) by XREAL_1:24,ABSVALUE:4;
      then y.i - |. (seq_p(d1)).i .| <= y.i -(-(seq_p(d1)).i) by XREAL_1:13;
      then y.i - |. (seq_p(d1)).i .| <= (seq_p(d1)).i + y.i;
      hence thesis by P1,SEQ_1:7,P32;
    end;
    hence thesis;
  end;
