
theorem u8:
for F being Field,
    E being FieldExtension of F
for G being non empty FinSequence of the carrier of Polynom-Ring E
st for i being Nat st i in dom G ex a being Element of F st G.i = rpoly(1,a)
holds (Product G) is Ppoly of F
proof
let F be Field, E be FieldExtension of F;
let G be non empty FinSequence of the carrier of Polynom-Ring E;
assume AS: for i being Nat st i in dom G
           ex a being Element of F st G.i = rpoly(1,a);
F is Subfield of E by FIELD_4:7; then
H1: the carrier of F c= the carrier of E by EC_PF_1:def 1;
defpred P[Nat] means
  for G being non empty FinSequence of Polynom-Ring E st len G = $1 &
  for i being Nat st i in dom G ex a being Element of F st G.i = rpoly(1,a)
  holds Product G is Ppoly of F;
IA: P[0];
IS: now let k be Nat;
    assume IV: P[k];
    now let G be non empty FinSequence of Polynom-Ring E;
      assume A: len G = k+1 & for i being Nat st i in dom G
                ex a being Element of F st G.i = rpoly(1,a);
    per cases;
    suppose S: k = 0;
      then 1 in Seg(len G) by A;
      then 1 in dom G by FINSEQ_1:def 3;
      then consider a being Element of F such that A0: G.1 = rpoly(1,a) by A;
      H: G = <*rpoly(1,a)*> by A0,A,S,FINSEQ_1:40;
      reconsider b = a as Element of E by H1;
      rpoly(1,a) = rpoly(1,b) by FIELD_4:21; then
      rpoly(1,a) is Element of the carrier of Polynom-Ring E
                                                          by POLYNOM3:def 10;
      then Product G = rpoly(1,a) by H,GROUP_4:9;
      hence Product G is Ppoly of F by RING_5:51;
      end;
    suppose S: k > 0;
      consider H being FinSequence, y being object such that
      B2: G = H^<*y*> by FINSEQ_1:46;
      B2a: rng H c= rng G by B2,FINSEQ_1:29;
      B2b: rng G c= the carrier of Polynom-Ring E by FINSEQ_1:def 4;
      then reconsider H as FinSequence of Polynom-Ring E
                                    by B2a,XBOOLE_1:1,FINSEQ_1:def 4;
      B3: len G = len H + len <*y*> by B2,FINSEQ_1:22
               .= len H + 1 by FINSEQ_1:39; then
      reconsider H as non empty FinSequence of Polynom-Ring E by S,A;
      reconsider q = Product H as Polynomial of E by POLYNOM3:def 10;
      C: dom H c= dom G by B2,FINSEQ_1:26;
      now let i be Nat;
         assume C0: i in dom H;
         then H.i = G.i by B2,FINSEQ_1:def 7;
         hence ex a being Element of F st H.i = rpoly(1,a) by C,C0,A;
         end;
      then reconsider q1 = Product H as Ppoly of F by B3,A,IV;
      rng<*y*> = {y} by FINSEQ_1:39;
      then G5: y in rng<*y*> by TARSKI:def 1;
      rng<*y*> c= rng G by B2,FINSEQ_1:30;
      then reconsider y as Element of Polynom-Ring E by G5,B2b;
      dom<*y*> = {1} by FINSEQ_1:2,FINSEQ_1:def 8;
      then 1 in dom<*y*> by TARSKI:def 1;
      then B6: G.(k+1) = <*y*>.1 by B2,B3,A,FINSEQ_1:def 7
                      .= y;
      dom G = Seg(k+1) by A,FINSEQ_1:def 3;
      then consider a being Element of F such that
      B9: y = rpoly(1,a) by A,B6,FINSEQ_1:4;
      reconsider b = a as Element of E by H1;
      reconsider y1 = rpoly(1,b) as Element of Polynom-Ring E
              by POLYNOM3:def 10;
      B8: rpoly(1,b) = rpoly(1,a) by FIELD_4:21;
      B7: rpoly(1,a) is Ppoly of F by RING_5:51;
      Product G = (Product H) * y by B2,GROUP_4:6
               .= q *' rpoly(1,b) by B8,B9,POLYNOM3:def 10
               .= q1 *' rpoly(1,a) by B8,FIELD_4:17;
      hence Product G is Ppoly of F by B7,RING_5:52;
      end;
     end;
    hence P[k+1];
    end;
I: for k being Nat holds P[k] from NAT_1:sch 2(IA,IS);
consider n being Nat such that J: len G = n;
thus thesis by I,J,AS;
end;
