
theorem lem23:
for F being ordered Field,
    P being Ordering of F
for f being FinSequence of the carrier of Polynom-Ring F
for p being non zero Polynomial of F
st p = Sum f &
   for i being Element of dom f, q being Polynomial of F
   st q = f.i holds deg q is even & LC q in P
holds deg p is even
proof
let F be ordered Field, P be Ordering of F;
let f be FinSequence of the carrier of Polynom-Ring F;
let p be non zero Polynomial of F;
assume AS: p = Sum f &
           for i being Element of dom f, q being Polynomial of F
           st q = f.i holds deg q is even & LC q in P;
defpred P[Nat] means
  for f being FinSequence of the carrier of Polynom-Ring F
  for p being non zero Polynomial of F
  st p = Sum f & len f = $1 &
     for i being Element of dom f, q being Polynomial of F
     st q = f.i holds deg q is even & LC q in P
  holds deg p is even & LC p in P;
IA: P[1]
    proof
    now let f be FinSequence of the carrier of Polynom-Ring F;
      let p be non zero Polynomial of F;
      assume A0: p = Sum f & len f = 1 &
        for i being Element of dom f, q being Polynomial of F
        st q = f.i holds deg q is even & LC q in P; then
      A1: f = <* f.1 *> by FINSEQ_1:40; then
      dom f = { 1 } by FINSEQ_1:38,FINSEQ_1:2; then
      reconsider e = 1 as Element of dom f by TARSKI:def 1;
      A2: f.e in rng f & rng f c= the carrier of Polynom-Ring F
          by A1,FUNCT_1:3; then
      reconsider q = f.e as Polynomial of F by POLYNOM3:def 10;
      deg q is even & LC q in P by A0;
      hence deg p is even & LC p in P by A0,A1,A2,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;
      let p be non zero Polynomial of F;
      assume AS3: p = Sum f & len f = k + 1 &
        for i being Element of dom f, q being Polynomial of F
        st q = f.i holds deg q is even & LC q in P; then
      f<> {}; then
      consider G being FinSequence, y being object such that
      A1: f = G^<*y*> by FINSEQ_1:46;
      rng G c= rng f by A1,FINSEQ_1:29; then
      reconsider G as FinSequence of the carrier of Polynom-Ring F
           by XBOOLE_1:1,FINSEQ_1:def 4;
      A5: rng f c= the carrier of Polynom-Ring F;
      A6: rng<*y*> c= rng f by A1,FINSEQ_1:30;
      rng<*y*> = {y} & y in {y} by FINSEQ_1:38,TARSKI:def 1; then
      reconsider y as Element of the carrier of Polynom-Ring F by A5,A6;
      A3: len f = len G + len<*y*> by A1,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 AS3,AS1;
      A4: Sum f = Sum G + Sum<*y*> by A1,RLVECT_1:41;
      reconsider qG = Sum G, qy = y as Polynomial of F by POLYNOM3:def 10;
      A7: deg qy is even & LC qy in P
          proof
          dom <*y*> = Seg 1 by FINSEQ_1:38; then
          1 in dom <*y*> by FINSEQ_1:3; then
          C: f.(len G + 1) = <*y*>.1 by A1,FINSEQ_1:def 7 .= y;
          1 <=len G + 1 & dom f = Seg(len G + 1) by NAT_1:11,A3,FINSEQ_1:def 3;
          then len G + 1 in dom f by FINSEQ_1:1;
          hence thesis by C,AS3;
          end;
      per cases;
      suppose qG is non zero; then
        reconsider qG as non zero Polynomial of F;
        C0: now let i be Element of dom G, q be Polynomial of F;
            assume B1: q = G.i;
            dom G c= dom f & i in dom G by A1,FINSEQ_1:26; then
            f.i = G.i & i in dom f by A1,FINSEQ_1:def 7;
            hence deg q is even & LC q in P by B1,AS3;
            end; then
        C1: deg qG is even & LC qG in P by A3,AS2,AS3;
            not LC qG in {0.F} by TARSKI:def 1; then
            C3: LC qG in P^+ by C1,XBOOLE_0:def 5;
        C9: LC qG + LC qy in P^+ + P & P^+ + P c= P^+ by C3,A7,lemP;
            not LC qG + LC qy in {0.F} by C9,XBOOLE_0:def 5; then
        C2: LC qG + LC qy <> 0.F by TARSKI:def 1; then
        C3: deg(qG + qy) = max(deg qG, deg qy) by lem23a;
        C4: p = Sum G + y by AS3,A4,RLVECT_1:44
             .= qG + qy by POLYNOM3:def 10;
        thus deg p is even
           proof
           per cases by C3,XXREAL_0:16;
           suppose deg(qG + qy) = deg qG;
             hence thesis by C4,C0,A3,AS2,AS3;
             end;
           suppose deg(qG + qy) = deg qy;
             hence thesis by C4,A7;
             end;
           end;
        LC(qG + qy) in P
          proof
          per cases by XXREAL_0:1;
          suppose deg qG > deg qy;
            hence thesis by C1,lem23d;
            end;
          suppose deg qG < deg qy;
            hence thesis by A7,lem23d;
            end;
          suppose deg qG = deg qy;
            then LC(qG + qy) = LC qG + LC qy by C2,lem23d;
            hence thesis by C9,XBOOLE_0:def 5;
            end;
          end;
        hence LC p in P by C4;
        end;
      suppose qG is zero; then
        qG = 0_.(F) by UPROOTS:def 5 .= 0.(Polynom-Ring F) by POLYNOM3:def 10;
        hence deg p is even & LC p in P by A7,A4,AS3,RLVECT_1:44;
        end;
      end;
   hence P[k+1];
   end;
I: for k being Nat st k >= 1 holds P[k] from NAT_1:sch 8(IA,IS);
p <> 0_.(F); then
p <> 0.(Polynom-Ring F) by POLYNOM3:def 10; then
f <> <*>(the carrier of Polynom-Ring F) by AS,RLVECT_1:43; then
len f >= 0 + 1 by INT_1:7;
hence deg p is even by AS,I;
end;
