
theorem lemma2u:
for F being Field,
    E being FieldExtension of F
for G1 being non empty FinSequence of Polynom-Ring F
st for i being Nat st i in dom G1
   ex a being Element of F st G1.i = rpoly(1,a)
for G2 being non empty FinSequence of Polynom-Ring E
st for i being Nat st i in dom G2
   ex a being Element of E st G2.i = rpoly(1,a)
holds Product G1 = Product G2 implies
for a being Element of E holds
(ex i being Nat st i in dom G1 & G1.i = rpoly(1,a)) iff
(ex i being Nat st i in dom G2 & G2.i = rpoly(1,a))
proof
let F be Field, E be FieldExtension of F;
let G1 be non empty FinSequence of Polynom-Ring F;
assume AS1: for i being Nat st i in dom G1
            ex a being Element of F st G1.i = rpoly(1,a);
let G2 be non empty FinSequence of Polynom-Ring E;
assume AS2: for i being Nat st i in dom G2
            ex a being Element of E st G2.i = rpoly(1,a);
assume AS3: Product G1 = Product G2;
H: Polynom-Ring F is Subring of Polynom-Ring E by FIELD_4:def 1;
reconsider p = Product G2 as Polynomial of E by POLYNOM3:def 10;
now let o be object;
  assume o in rng G1; then
  consider i being object such that
  H1: i in dom G1 & G1.i = o by FUNCT_1:def 3;
  reconsider i as Nat by H1;
  consider a being Element of F such that H2: G1.i = rpoly(1,a) by H1,AS1;
  F is Subring of E by FIELD_4:def 1;
  then the carrier of F c= the carrier of E by C0SP1:def 3;
  then reconsider b = a as Element of E;
  rpoly(1,a) = rpoly(1,b) by FIELD_4:21;
  hence o in the carrier of Polynom-Ring E by H1,H2,POLYNOM3:def 10;
  end;
then rng G1 c= the carrier of Polynom-Ring E; then
reconsider G1a = G1 as non empty FinSequence of Polynom-Ring E
  by FINSEQ_1:def 4;
AS4: now let i be Nat;
     assume i in dom G1a; then
     consider a being Element of F such that L: G1.i = rpoly(1,a) by AS1;
     F is Subring of E by FIELD_4:def 1;
     then the carrier of F c= the carrier of E by C0SP1:def 3;
     then reconsider b = a as Element of E;
     rpoly(1,a) = rpoly(1,b) by FIELD_4:21;
     hence ex a being Element of E st G1a.i = rpoly(1,a) by L;
     end;
Product G1a = Product G2 by AS3,H,u5;
hence thesis by AS2,AS4,lemma2z;
end;
