
theorem TLNEG36:
  for a be Real st 0 < a holds
  for k be Nat, d be nonnegative-yielding XFinSequence of REAL
  st len d = k holds
  ex N be Nat st
  for x be Nat st N <=x holds
  for i be Nat st i in dom d
  holds (d.i) * (x to_power i)*k <= a* (x to_power k)
  proof
    let a be Real;
    assume AS: 0 < a;
    let k be Nat, d be nonnegative-yielding XFinSequence of REAL;
    assume
    A1: len d = k;
    A2: 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 in rng d by FUNCT_1:3;
      hence thesis by PARTFUN3:def 4;
    end;
    per cases;
    suppose P0:k= 0;
      set N = 0;
      take N;
      let x be Nat;
      assume N <=x;
      thus for i be Nat st i in dom d holds (d.i) * (x to_power i)*k
      <= a* (x to_power k) by P0,A1;
    end;
    suppose K1P:k <> 0; then
      0 <= k-1 by XREAL_1:48,NAT_1:14; then
      reconsider k1=k-1 as Nat by INT_1:3,ORDINAL1:def 12;
      consider M be Real such that
      Q1: 0 <= M & for i be Nat st i in dom d holds d.i <= M by TLNEG42,A2;
      set N = [/ M*k/a \] + 2;
      MM1:M*k/a <= [/ M*k/a \] by INT_1:def 7;
      Q20P:0 <= M*k/a by AS,Q1; then
      Q20: 1 + 0 < 2 + [/ M*k/a \] by XREAL_1:8,MM1;
      N in NAT by INT_1:3,MM1,Q20P; then
      reconsider N as Nat;
      [/ M*k/a \] + 1 + 0 < [/ M*k/a \] + 1 + 1 by XREAL_1:8; then
      M*k/a < N by XXREAL_0:2,INT_1:32; then
      M*k/a *a <= N *a by AS,XREAL_1:64; then
      Q2: M*k <= a*N & 1 < N by XCMPLX_1:87,AS,Q20;
      take N;
      thus for x be Nat st N <= x holds for i be Nat st i in dom d
      holds (d.i) * (x to_power i)*k <= a* (x to_power k)
      proof
        let x be Nat;
        assume Q3: N <=x;
        let i be Nat;
        assume Q4: i in dom d; then
        (d.i)*k <= M*k by XREAL_1:64,Q1; then
        Q5: (d.i)*k*(x to_power i) <= M*k *(x to_power i) by XREAL_1:64;
        i in Segm k by Q4,A1; then
        i < k1+1 by NAT_1:44; then
        Y1: i <= k1 by NAT_1:13;
        Y2: 1 < x by Q3,XXREAL_0:2,Q20;
        X1: x to_power i <= x to_power k1
        proof
          per cases;
          suppose i = k1;
            hence x to_power i <= x to_power k1;
          end;
          suppose i <> k1; then
            i < k1 by Y1,XXREAL_0:1;
            hence x to_power i <= x to_power k1 by Y2,POWER:39;
          end;
        end;
        N1:x*(x to_power k1) = x to_power k
        proof
          per cases;
          suppose x = 0;
            hence x*(x to_power k1) = x to_power k by K1P,POWER:def 2;
          end;
          suppose XX1: x <> 0;
            x = x to_power 1 by POWER:25;
            hence x*(x to_power k1) = x to_power (1 +k1) by XX1,POWER:27
            .= x to_power k;
          end;
        end;
        (M*k) *(x to_power i) <=(M*k) * (x to_power k1) by X1,XREAL_1:64,Q1;
        then
        Q7: (d.i)*k*(x to_power i)
        <=M*k * (x to_power k1) by XXREAL_0:2,Q5;
        M*k * (x to_power k1) <= a*N* (x to_power k1) by Q2,XREAL_1:64;
        then
        Q8: (d.i)*k*(x to_power i)
        <= a*N* (x to_power k1) by XXREAL_0:2,Q7;
        a*N <= a*x by XREAL_1:64,AS,Q3; then
        a*N* (x to_power k1) <= a*x*(x to_power k1) by XREAL_1:64;
        hence (d.i)*(x to_power i)*k <= a*(x to_power k) by XXREAL_0:2,Q8,N1;
      end;
    end;
  end;
