
theorem lem24:
for F being Field
for f being FinSequence of the carrier of Polynom-Ring F
for p being non zero Polynomial of F st p = Sum f
for g being FinSequence of F
for n being Nat
st for i being Element of dom f, q being Polynomial of F
   st q = f.i holds deg q <= n
holds deg p <= n
proof
let F be Field, f be FinSequence of the carrier of Polynom-Ring F;
let p be non zero Polynomial of F;
assume AS1: p = Sum f;
let g be FinSequence of F, n be Nat;
assume AS2: for i being Element of dom f, q being Polynomial of F
            st q = f.i holds deg q <= n;
defpred P[Nat] means
  for f being FinSequence of the carrier of Polynom-Ring F
  for p being Polynomial of F st len f = $1 & p = Sum f
  for g being FinSequence of F
  for n being Nat
  st for i being Element of dom f, q being Polynomial of F
     st q = f.i holds deg q <= n
  holds deg p <= n;
IA: P[1]
    proof
    now let f be FinSequence of the carrier of Polynom-Ring F,
            p be Polynomial of F;
      assume A1: len f = 1 & p = Sum f;
      let g be FinSequence of F, n be Nat;
      assume A2: for i being Element of dom f, q being Polynomial of F
         st q = f.i holds deg q <= n;
      A3: f = <*f.1*> by A1,FINSEQ_1:40; then
      dom f = Seg 1 by FINSEQ_1:38; then
      A4: 1 in dom f by FINSEQ_1:2,TARSKI:def 1; then
      A5: f.1 in rng f by FUNCT_1:3; then
      reconsider q = f.1 as Polynomial of F by A1,A3,RLVECT_1:44;
      thus deg p <= n by A5,A4,A2,A1,A3,RLVECT_1:44;
      end;
    hence thesis;
    end;
IS: now let k be Nat;
    assume AS1: k >= 1;
    assume AS2: P[k];
    now let f be FinSequence of the carrier of Polynom-Ring F,
            p be Polynomial of F;
      assume A1: len f = k + 1 & p = Sum f;
      let g be FinSequence of F, n be Nat;
      assume A2: for i being Element of dom f, q being Polynomial of F
         st q = f.i holds deg q <= n;
      f <> {} by A1; then
      consider G being FinSequence, y being object such that
      A3: f = G^<*y*> by FINSEQ_1:46;
      rng G c= rng f by A3,FINSEQ_1:29; then
      reconsider G as FinSequence of the carrier of Polynom-Ring F
           by XBOOLE_1:1,FINSEQ_1:def 4;
      A4: len f = len G + len<*y*> by A3,FINSEQ_1:22
               .= len G + 1 by FINSEQ_1:39; then
      reconsider G as non empty FinSequence of the carrier of Polynom-Ring F
          by AS1,A1;
      reconsider r = Sum G as Polynomial of F by POLYNOM3:def 10;
      now let i be Element of dom G, q be Polynomial of F;
        assume B1: q = G.i;
        B2: i in dom G & dom G c= dom f by A3,FINSEQ_1:26;
        G.i = f.i by A3,FINSEQ_1:def 7;
        hence deg q <= n by B1,B2,A2;
        end; then
      A5: deg r <= n by A1,A4,AS2;
      rng<*y*> = {y} by FINSEQ_1:39; then
      A6: y in rng<*y*> by TARSKI:def 1;
      rng<*y*> c= rng f by A3,FINSEQ_1:30; then
      consider u being object such that
      A7: u in dom f & f.u = y by A6,FUNCT_1:def 3;
      reconsider u as Element of NAT by A7;
      f.u in rng f & rng f c= the carrier of Polynom-Ring F
         by A7,FUNCT_1:3; then
      reconsider y as Element of the carrier of Polynom-Ring F by A7;
      reconsider s = y as Polynomial of F;
      A8: Sum <*y*> = s by RLVECT_1:44;
      dom f = Seg(len G + 1) & 1 <= len G + 1
               by A4,FINSEQ_1:def 3,NAT_1:11; then
      A9: len G + 1 in dom f by FINSEQ_1:1;
      f.(len G + 1) = y by A3,FINSEQ_1:42; then
      A10: deg s <= n by A2,A9;
      p = Sum G + Sum <*y*> by A1,A3,RLVECT_1:41
       .= r + s by A8,POLYNOM3:def 10; then
      A11: deg p <= max(deg r, deg s) by HURWITZ:22;
      max(deg r, deg s) <= n by A5,A10,XXREAL_0:28;
      hence deg p <= n by A11,XXREAL_0:2;
      end;
    hence P[k+1];
    end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(IA,IS);
now assume f = {};
  then f = <*>(the carrier of Polynom-Ring F);
  then Sum f = 0.(Polynom-Ring F) & p <> 0_.(F) by RLVECT_1:43;
  hence contradiction by AS1,POLYNOM3:def 10;
  end;
then len f >= 0 + 1 by INT_1:7;
hence thesis by I,AS1,AS2;
end;
