
theorem lemNor1b:
for F being Field,
    E being FieldExtension of F
for G being non empty FinSequence of the carrier of Polynom-Ring F
for q being Polynomial of F st q = Product G
for a being Element of E
st ex i being Element of dom G,
      p being Polynomial of F st p = G.i & Ext_eval(p,a) = 0.E
holds Ext_eval(q,a) = 0.E
proof
let F be Field, E be FieldExtension of F;
let G be non empty FinSequence of the carrier of Polynom-Ring F;
let q be Polynomial of F;
assume A: q = Product G;
let a be Element of E;
assume ex i being Element of dom G,
          p being Polynomial of F st p = G.i & Ext_eval(p,a) = 0.E; then
consider i being Element of dom G, p being Polynomial of F such that
B: p = G.i & Ext_eval(p,a) = 0.E;
H: F is Subring of E by FIELD_4:def 1;
   dom G = Seg(len G) by FINSEQ_1:def 3; then
   1 <= i & i <= len G by FINSEQ_1:1; then
   G = (G|(i-'1)) ^ <*G.i*> ^ (G/^i) by FINSEQ_5:84; then
D: Product G
 = Product(G|(i-'1)^<*G.i*>) * Product((G/^i)) by GROUP_4:5
.= (Product(G|(i-'1)) * Product(<*G.i*>)) * Product(G/^i) by GROUP_4:5
.= (Product(<*G.i*>) * Product(G|(i-'1))) * Product(G/^i) by GROUP_1:def 12
.= Product(<*G.i*>) *(Product(G|(i-'1)) * Product(G/^i)) by GROUP_1:def 3;
reconsider r1 = Product(G|(i-'1)),
           r2 = Product(G/^i) as Polynomial of F by POLYNOM3:def 10;
I: G.i = G/.i by PARTFUN1:def 6; then
E: Product(<*G/.i*>) = p by B,GROUP_4:9;
r1 *' r2 = Product(G|(i-'1)) * Product(G/^i) by POLYNOM3:def 10; then
q = p *'(r1 *' r2) by A,D,E,I,POLYNOM3:def 10;
hence Ext_eval(q,a)
         = 0.E * Ext_eval(r1*'r2,a) by B,H,ALGNUM_1:20
        .= 0.E;
end;
