
theorem lemma7a:
for F being Field,
    p being Element of the carrier of Polynom-Ring F
for E being FieldExtension of F,
    U being E-extending FieldExtension of F
for a being Element of E, b being Element of U st a = b
holds Ext_eval(p,a) = Ext_eval(p,b)
proof
let F be Field,
    p be Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F,
    U be E-extending FieldExtension of F;
let a be Element of E, b be Element of U;
assume AS2: a = b;
H1: E is Subring of U by FIELD_4:def 1; then
H2: the carrier of E c= the carrier of U by C0SP1:def 3;
F is Subring of E by FIELD_4:def 1; then
H4: the carrier of F c= the carrier of E by C0SP1:def 3;
consider Fp being FinSequence of E 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),E) * (power E).(a,n-'1) by ALGNUM_1:def 1;
consider Fq being FinSequence of U such that
B: Ext_eval(p,b) = Sum Fq & len Fq = len p &
   for n being Element of NAT st n in dom Fq holds
   Fq.n = In(p.(n-'1),U) * (power U).(b,n-'1) by ALGNUM_1:def 1;
C: now let n be Element of NAT;
   thus (power U).(b,n-'1) = (power U).(In(b,U),n-'1)
                          .= In((power E).(a,n-'1),U) by AS2,H1,ALGNUM_1:7
                          .= (power E).(a,n-'1) by H2,SUBSET_1:def 8;
   end;
D: dom Fp = Seg(len Fq) by A,B,FINSEQ_1:def 3 .= dom Fq by FINSEQ_1:def 3;
E: now let n be Element of NAT;
   p.(n-'1) in U by H2,H4;
   hence In(p.(n-'1),U) = p.(n-'1) by SUBSET_1:def 8
                       .= In(p.(n-'1),E) by H4,SUBSET_1:def 8;
   end;
H: for n being Element of NAT holds [In(p.(n-'1),E),(power E).(a,n-'1)]
                 in [:the carrier of E,the carrier of E:] by ZFMISC_1:def 2;
now let n be Nat;
  assume F: n in dom Fq;
  G: n is Element of NAT by ORDINAL1:def 12;
  thus Fq.n = In(p.(n-'1),U) * (power U).(b,n-'1) by B,F
           .= (the multF of U).(In(p.(n-'1),U),(power E).(a,n-'1)) by C,G
           .= (the multF of U).(In(p.(n-'1),E),(power E).(a,n-'1)) by E,G
           .= ((the multF of U)||(the carrier of E)).
                       (In(p.(n-'1),E),(power E).(a,n-'1)) by G,H,FUNCT_1:49
           .= In(p.(n-'1),E) * (power E).(a,n-'1) by H1,C0SP1:def 3
           .= Fp.n by A,D,F;
  end;
then Fp = Fq by D;
hence thesis by H1,A,B,FIELD_4:2;
end;
