
theorem TLNEG35:
  for k be Nat,
  d be XFinSequence of REAL,
  a be Real,
  y be Real_Sequence
  st 0 < a
  & len d = k
  & for x be Nat holds y.x = a * (x to_power k)
  holds
  ex N be Nat st
  for x be Nat st N <= x holds |.(seq_p(d)).x .| <= y.x
  proof
    let k be Nat, d be XFinSequence of REAL,
    a be Real, y be Real_Sequence;
    assume that
    A1: 0 < a and
    A2:len d = k and
    A3:for x be Nat holds y.x = a* (x to_power k);
    per cases;
    suppose K1: k = 0;
      set N = 0;
      take N;
      thus for x be Nat st N <=x holds |.(seq_p(d)).x .| <= y.x
      proof
        let x be Nat;
        assume N <=x;
        D2: |.(seq_p(d)).x .| = 0 by COMPLEX1:44,A2,K1,TLNEG1;
        y.x = a* (x to_power k) by A3;
        hence thesis by D2,A1;
      end;
    end;
    suppose B4: k <> 0;
      reconsider c = |. d .| as XFinSequence of REAL;
      len c = k by A2,VALUED_1:def 11;
      then consider N be Nat such that
      A4: for x be Nat st N <=x holds
      for i be Nat st i in dom c holds ( (c.i) * x to_power i ) *k
      <= a* (x to_power k) by A1,TLNEG36;
      take N;
      thus for x be Nat st N <=x holds |.(seq_p(d)).x .| <= y.x
      proof
        let x be Nat;
        assume A0:N <=x;
        NN0:dom (c (#) seq_a^(x,1,0)) = dom c by LMXFIN1
        .= dom d by VALUED_1:def 11;
        P1: (seq_p(c)).x = Sum(c (#) seq_a^(x,1,0)) by defseqp;
        for i be Nat st i in dom (c (#) seq_a^(x,1,0)) holds
        (c (#) seq_a^(x,1,0)).i <= (y.x) / k
        proof
          let i be Nat;
          assume i in dom (c (#) seq_a^(x,1,0)); then
          X5: i in dom c by LMXFIN1; then
          (c.i) * (x to_power i)*k/k <= a* (x to_power k)/k
          by XREAL_1:72,A0,A4; then
          (c.i) * x to_power i <= a* (x to_power k)/k by B4,XCMPLX_1:89;
          then
          (c.i) * x to_power i <= (y.x) / k by A3;
          hence (c (#) seq_a^(x,1,0)).i <= (y.x) / k by X5,LMXFIN2;
        end; then
        Sum (c (#) seq_a^(x,1,0)) <= ((y.x)/k ) *len (c (#) seq_a^(x,1,0))
        by AFINSQ_2:59; then
        P6: ( seq_p(c) ).x <= y.x by P1,NN0,A2,B4,XCMPLX_1:87;
        |.(seq_p(d)).x .| <= ( seq_p(c) ).x by TLNEG41;
        hence |.(seq_p(d)).x .| <= y.x by XXREAL_0:2,P6;
      end;
    end;
  end;
