
theorem ID:
for F being Field
for E being FieldExtension of F
for a being F-algebraic Element of E
for f,g being F-fixing Automorphism of FAdj(F,{a}) st f.a = g.a holds f = g
proof
let F be Field, E be FieldExtension of F, a be F-algebraic Element of E;
let f,g be F-fixing Automorphism of FAdj(F,{a});
assume AS: f.a = g.a;
A: E is (Polynom-Ring F)-homomorphic FieldExtension of F; then
B: the carrier of FAdj(F,{a})
    = the carrier of RAdj(F,{a}) by FIELD_6:56
   .= the set of all Ext_eval(p,a) where p is Polynomial of F by A,FIELD_6:45;
defpred P[Nat] means
  for p being Polynomial of F
  st deg p = $1 holds f.Ext_eval(p,a) = g.Ext_eval(p,a);
IA: now let p be non zero Polynomial of F;
    F is Subfield of FAdj(F,{a}) by FIELD_4:7; then
    the carrier of F c= the carrier of FAdj(F,{a}) by EC_PF_1:def 1; then
    reconsider b = LC p as Element of FAdj(F,{a});
    a in {a} & {a} is Subset of FAdj(F,{a}) by TARSKI:def 1,FIELD_6:35; then
    reconsider a1 = a as Element of FAdj(F,{a});
    A: Ext_eval(LM p,a1) = b * (a1|^(deg p)) by FIELD_6:29;
    E is FAdj(F,{a})-extending &
    LM p is Element of the carrier of Polynom-Ring F
       by FIELD_4:7,POLYNOM3:def 10; then
    B: Ext_eval(LM p,a) = Ext_eval(LM p,a1) by FIELD_6:11;
    id FAdj(F,{a}) is additive multiplicative unity-preserving; then
    C: FAdj(F,{a}) is FAdj(F,{a})-homomorphic by RING_2:def 4;
    thus f.Ext_eval(LM p,a)
        = f.b * f.(a1|^(deg p)) by A,B,GROUP_6:def 6
       .= b * f.(a1|^(deg p)) by FIELD_8:def 2
       .= b * (f.a1)|^(deg p) by C,lemID
       .= b * g.(a1|^(deg p)) by AS,C,lemID
       .= g.b * g.(a1|^(deg p)) by FIELD_8:def 2
       .= g.Ext_eval(LM p,a) by A,B,GROUP_6:def 6;
    end;
IS: now let k be Nat;
    assume IV: for n being Nat st n < k holds P[n];
    now let p be Polynomial of F;
      assume A0: deg p = k; then
      H1: p <> 0_.(F) by HURWITZ:20; then
      consider q being Polynomial of F such that
      A1: len q < len p & p = q + LM(p) &
          for n being Element of NAT st n < len p-1 holds q.n = p.n
          by POLYNOM4:16,POLYNOM4:5;
      H2: p is non zero by H1,UPROOTS:def 5;
      per cases;
      suppose q = 0_.(F);
        hence f.Ext_eval(p,a) = g.Ext_eval(p,a) by A1,H2,IA;
        end;
      suppose q <> 0_.(F);
        then reconsider degq = deg q as Nat by FIELD_1:1;
        deg p = len p - 1 by HURWITZ:def 2; then
        H: len q < k + 1 by A0,A1;
        deg q = len q - 1 by HURWITZ:def 2; then
        len q = deg q + 1; then
        degq < k by XREAL_1:6,H; then
        A3: f.Ext_eval(q,a) = g.Ext_eval(q,a) by IV;
        A4: f.Ext_eval(LM p,a) = g.Ext_eval(LM p,a) by H2,IA;
        A5: F is Subring of E by FIELD_4:def 1;
        Ext_eval(q,a) in
            the set of all Ext_eval(p,a) where p is Polynomial of F &
        Ext_eval(LM p,a) in
            the set of all Ext_eval(p,a) where p is Polynomial of F; then
        reconsider u = Ext_eval(q,a), v = Ext_eval(LM p,a)
                           as Element of the carrier of FAdj(F,{a}) by B;
        A7: Ext_eval(q + LM p,a) = Ext_eval(q,a) + Ext_eval(LM p,a)
            by A5,ALGNUM_1:15;
        A8: FAdj(F,{a}) is Subring of E by FIELD_5:12;
        thus f.Ext_eval(p,a)
           = f.(u + v) by A1,A7,A8,FIELD_6:15
          .= f.u + f.v by VECTSP_1:def 20
          .= g.(u + v) by A3,A4,VECTSP_1:def 20
          .= g.Ext_eval(p,a) by A1,A7,A8,FIELD_6:15;
        end;
      end;
    hence P[k];
    end;
I: for k being Nat holds P[k] from NAT_1:sch 4(IS);
now let o be object;
  assume o in the carrier of FAdj(F,{a}); then
  consider p being Polynomial of F such that
  C: o = Ext_eval(p,a) by B;
  per cases;
  suppose p = 0_.(F); then
    Ext_eval(p,a) = 0.E by ALGNUM_1:13
                 .= 0.(FAdj(F,{a})) by FIELD_6:def 6;
    hence f.o = g.o by C;
    end;
  suppose p <> 0_.(F); then
    deg p is natural by FIELD_1:1; then
    deg p in NAT by ORDINAL1:def 12;
    hence f.o = g.o by C,I;
    end;
  end;
hence thesis;
end;
