
theorem lemphi3:
for n being Nat
for F being Field,
    E being FieldExtension of F
for x being Function of n,E holds F is Subring of Image_hom_Ext_eval(x,F)
proof
let n be Nat, F be Field, E be FieldExtension of F, x be Function of n,E;
set R = Image_hom_Ext_eval(x,F), f = hom_Ext_eval(x,F);
U: F is Subring of E by FIELD_4:def 1; then
Y: the carrier of F c= the carrier of E by C0SP1:def 3;
Z: now let o be object;
   assume o in the carrier of F;
   then reconsider b = o as Element of F;
   reconsider c = b as Element of E by Y;
   reconsider p = b|(n,F) as Element of the carrier of Polynom-Ring(n,F)
      by POLYNOM1:def 11;
   reconsider q = c|(n,E) as Element of the carrier of Polynom-Ring(n,E)
      by POLYNOM1:def 11;
   Ext_eval(b|(n,F),x) = eval(c|(n,E),x) by field427,field426
                      .= b by POLYNOM7:25;
   then b in the set of all Ext_eval(p,x) where p is Polynomial of n,F;
   hence o in rng hom_Ext_eval(x,F) by lemphi1;
   end; then
ZZ: the carrier of F c= rng f;
now let o be object;
   assume o in the carrier of F;
   then o in rng hom_Ext_eval(x,F) by Z;
   hence o in the carrier of R by defIm;
   end;
then A: the carrier of F c= the carrier of R;
set adF = the addF of F, adR = the addF of R;
H1c: dom(the addF of E) = [:the carrier of E, the carrier of E:]
     by FUNCT_2:def 1;
adR = (the addF of E)||(rng f) by defIm
   .= (the addF of E)|[:rng f,rng f:]; then
H1b: dom adR = [:the carrier of E, the carrier of E:] /\ [:rng f,rng f:]
     by H1c,RELAT_1:61;
H1: dom(adR||the carrier of F)
  = dom(adR) /\ [:the carrier of F, the carrier of F:] by RELAT_1:61
 .= [:the carrier of E, the carrier of E:] /\
    ([:rng f,rng f:] /\ [:the carrier of F, the carrier of F:])
    by H1b,XBOOLE_1:16
 .= [:the carrier of E, the carrier of E:] /\
    [:the carrier of F, the carrier of F:] by ZZ,ZFMISC_1:96,XBOOLE_1:28
 .= [:the carrier of F, the carrier of F:] by Y,ZFMISC_1:96,XBOOLE_1:28
 .= dom adF by FUNCT_2:def 1;
now let o be object;
  assume AS: o in dom adF;
  then consider a,b being object such that
  B1: a in the carrier of F &
      b in the carrier of F & o = [a,b] by ZFMISC_1:def 2;
  reconsider a,b as Element of F by B1;
  a in rng f & b in rng f by Z; then
  B3: o in [:rng f, rng f:] by B1,ZFMISC_1:def 2;
  B5: the addF of F = (the addF of E)||(the carrier of F) by C0SP1:def 3,U;
  thus adF.o = (the addF of E).o by AS,B5,FUNCT_1:49
            .= ((the addF of E)||(rng f)).o by B3,FUNCT_1:49
            .= adR.o by defIm
            .= (adR|[:the carrier of F, the carrier of F:]).o
               by AS,H1,FUNCT_1:47;
  end;
then B: the addF of F = (the addF of R)||the carrier of F by H1;
set muF = the multF of F, muR = the multF of R;
H1c: dom(the multF of E) = [:the carrier of E, the carrier of E:]
     by FUNCT_2:def 1;
muR = (the multF of E)||(rng f) by defIm
   .= (the multF of E)|[:rng f,rng f:]; then
H1b: dom muR = [:the carrier of E, the carrier of E:] /\ [:rng f,rng f:]
     by H1c,RELAT_1:61;
H1: dom(muR||the carrier of F)
  = dom(muR) /\ [:the carrier of F, the carrier of F:] by RELAT_1:61
 .= [:the carrier of E, the carrier of E:] /\
    ([:rng f,rng f:] /\ [:the carrier of F, the carrier of F:])
    by H1b,XBOOLE_1:16
 .= [:the carrier of E, the carrier of E:] /\
    [:the carrier of F, the carrier of F:] by ZZ,ZFMISC_1:96,XBOOLE_1:28
 .= [:the carrier of F, the carrier of F:] by Y,ZFMISC_1:96,XBOOLE_1:28
 .= dom muF by FUNCT_2:def 1;
now let o be object;
  assume AS: o in dom muF;
  then consider a,b being object such that
  B1: a in the carrier of F &
      b in the carrier of F & o = [a,b] by ZFMISC_1:def 2;
  reconsider a,b as Element of F by B1;
  a in rng f & b in rng f by Z; then
  B3: o in [:rng f, rng f:] by B1,ZFMISC_1:def 2;
  B5: the multF of F=(the multF of E)||(the carrier of F) by C0SP1:def 3,U;
  thus muF.o = (the multF of E).o by AS,B5,FUNCT_1:49
            .= ((the multF of E)||(rng f)).o by B3,FUNCT_1:49
            .= muR.o by defIm
            .= (muR|[:the carrier of F, the carrier of F:]).o
               by AS,H1,FUNCT_1:47;
  end;
then C: the multF of F = (the multF of R)||the carrier of F by H1;
D: 0.F = 0.E by U,C0SP1:def 3 .= 0.R by defIm;
1.F = 1.E by U,C0SP1:def 3 .= 1.R by defIm;
hence thesis by A,B,C,D,C0SP1:def 3;
end;
