
theorem lemphi4:
for F being Field,
    E being (Polynom-Ring F)-homomorphic FieldExtension of F
for a being Element of E holds RAdj(F,{a}) = Image hom_Ext_eval(a,F)
proof
let F be Field, E be (Polynom-Ring F)-homomorphic FieldExtension of F;
let a be Element of E;
A0: F is Subring of E by FIELD_4:def 1;
set R = Image(hom_Ext_eval(a,F)), S = RAdj(F,{a}), f = hom_Ext_eval(a,F);
now let o be object;
  assume o in {a}; then
  A1: o = a by TARSKI:def 1;
  reconsider p = <%0.F,1.F%> as Element of the carrier of Polynom-Ring F
    by POLYNOM3:def 10;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
    by FIELD_4:10;
  then reconsider q = p as Element of the carrier of Polynom-Ring E;
  A2: dom f = the carrier of Polynom-Ring F by FUNCT_2:def 1;
  0.E = 0.F & 1.E = 1.F by A0,C0SP1:def 3;
  then A3: q = <%0.E,1.E%> by A0,pr20;
  f.p = Ext_eval(p,a) by ALGNUM_1:def 11
     .= eval(q,a) by FIELD_4:26
     .= 0.E + 1.E * a by A3,POLYNOM5:44
     .= a;
  then a in rng f by A2,FUNCT_1:def 3;
  hence o in the carrier of R by A1,RING_2:def 6;
  end; then
A: {a} is Subset of the carrier of R by TARSKI:def 3;
F is Subring of R by lemphi3; then
S is Subring of R by A,RAsub2; then
the carrier of RAdj(F,{a}) c= the carrier of Image f by C0SP1:def 3; then
V: the carrier of RAdj(F,{a}) c= rng hom_Ext_eval(a,F) by RING_2:def 6;
rng hom_Ext_eval(a,F) c= the carrier of RAdj(F,{a}) by lemphi2;
then the carrier of RAdj(F,{a}) = rng f by V,TARSKI:2
  .= the carrier of Image f by RING_2:def 6;
hence thesis by RE1;
end;
