
theorem lemma7b:
for F being Field,
    p being Element of the carrier of Polynom-Ring F
for E being FieldExtension of F,
    q being Element of the carrier of Polynom-Ring E st q = p
for U being E-extending FieldExtension of F
for a being Element of U
holds Ext_eval(q,a) = Ext_eval(p,a)
proof
let F be Field,
    p be Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F,
    q being Element of the carrier of Polynom-Ring E;
assume AS1: q = p;
let U be E-extending FieldExtension of F;
let a be Element of U;
consider Fp being FinSequence of U such that
A: Ext_eval(p,a) = Sum Fp & len Fp = len p &
   for n being Element of NAT st n in dom Fp holds
   Fp.n = In(p.(n-'1),U) * (power U).(a,n-'1) by ALGNUM_1:def 1;
consider Fq being FinSequence of U such that
B: Ext_eval(q,a) = Sum Fq & len Fq = len q &
   for n being Element of NAT st n in dom Fq holds
   Fq.n = In(q.(n-'1),U) * (power U).(a,n-'1) by ALGNUM_1:def 1;
C: len p - 1 = deg p by HURWITZ:def 2
            .= deg q by AS1,FIELD_4:20
            .= len q - 1 by HURWITZ:def 2;
D: dom Fp = Seg(len Fq) by A,B,C,FINSEQ_1:def 3
         .= dom Fq by FINSEQ_1:def 3;
now let n be Nat;
  assume F: n in dom Fq;
  hence Fq.n = In(p.(n-'1),U) * (power U).(a,n-'1) by B,AS1
            .= Fp.n by A,D,F;
  end;
then Fp = Fq by D;
hence thesis by A,B;
end;
