
theorem lemh1:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for T1 being finite F-algebraic Subset of E,
    T2 being Subset of K st T1 = T2 holds FAdj(F,T1) = FAdj(F,T2)
proof
let F be Field, E be FieldExtension of F,
    K be E-extending FieldExtension of F;
let T1 be finite F-algebraic Subset of E, T2 be Subset of K;
assume AS: T1 = T2;
defpred P[Nat] means
  for T1 being finite F-algebraic Subset of E, T2 be Subset of K
  st card T1 = $1 & T1 = T2 holds FAdj(F,T1) = FAdj(F,T2);
A: P[0]
   proof
   now let T1 be finite F-algebraic Subset of E, T2 be Subset of K;
   assume card T1 = 0 & T1 = T2; then
   T1 = {} & T2 = {}; then
   A1: T1 is Subset of F & T2 is Subset of F by XBOOLE_1:2; then
   the doubleLoopStr of FAdj(F,T1)
        = the doubleLoopStr of F by FIELD_7:def 1,FIELD_7:3
       .= the doubleLoopStr of FAdj(F,T2) by A1,FIELD_7:def 1,FIELD_7:3;
   hence FAdj(F,T1) = FAdj(F,T2);
   end;
   hence thesis;
   end;
B: now let k be Nat;
   assume B0: P[k];
   B1: E is Subfield of K by FIELD_4:7; then
   B2: the carrier of E c= the carrier of K by EC_PF_1:def 1;
   now let T1 be finite F-algebraic Subset of E, T2 be Subset of K;
     assume C0: card T1 = k+1 & T1 = T2; then
     reconsider U1 = T1 as non empty finite Subset of E;
     set a = the Element of U1;
     reconsider V1 = T1 \ {a} as finite F-algebraic Subset of E;
     reconsider V2 = T2 \ {a} as Subset of K;
         now let o be object;
           assume o in {a}; then
           o = a by TARSKI:def 1;
           hence o in T1;
           end; then
         {a} c= T1; then
     C1: T1 = V1 \/ {a} by XBOOLE_1:45;
         a in {a} by TARSKI:def 1; then
         not a in V1 by XBOOLE_0:def 5; then
     C2: card T1 = card V1 + 1 by C1,CARD_2:41;
         now let o be object;
           assume o in {a}; then
           o = a by TARSKI:def 1;
           hence o in T2 by C0;
           end; then
         {a} c= T2; then
     C3: T2 = V2 \/ {a} by XBOOLE_1:45;
     C4: FAdj(F,V1) = FAdj(F,V2) by B0,C0,C2;
     reconsider b = a as Element of K by B2;
     consider p being non zero Polynomial of F such that
     B: Ext_eval(p,a) = 0.E by FIELD_6:43;
     reconsider p as non zero Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
     Ext_eval(p,b) = 0.E by B,FIELD_6:11
                  .= 0.K by B1,EC_PF_1:def 1; then
     reconsider b as F-algebraic Element of K by FIELD_6:43;
     reconsider E1 = E as F-extending FieldExtension of FAdj(F,V1)
         by FIELD_4:7;
     reconsider E2 = K as F-extending FieldExtension of FAdj(F,V2)
         by FIELD_4:7;
     consider p being non zero Polynomial of F such that
     B: Ext_eval(p,a) = 0.E by FIELD_6:43;
     reconsider p as non zero Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
     the carrier of Polynom-Ring F
        c= the carrier of Polynom-Ring FAdj(F,V1) by FIELD_4:10; then
     reconsider q = p as Element of the carrier of Polynom-Ring FAdj(F,V1);
     C: Polynom-Ring F is Subring of Polynom-Ring FAdj(F,V1) by FIELD_4:def 1;
     now assume q is zero;
       then q = 0_.FAdj(F,V1) by UPROOTS:def 5
             .= 0.(Polynom-Ring FAdj(F,V1)) by POLYNOM3:def 10
             .= 0.(Polynom-Ring F) by C,C0SP1:def 3
             .= 0_.(F) by POLYNOM3:def 10;
       hence contradiction;
       end; then
     reconsider q as non zero Element of
                                  the carrier of Polynom-Ring FAdj(F,V1);
     Ext_eval(q,a) = 0.E1 by B,FIELD_7:15; then
     reconsider a1 = a as FAdj(F,V1)-algebraic Element of E1 by FIELD_6:43;
     consider p being non zero Polynomial of F such that
     B: Ext_eval(p,b) = 0.K by FIELD_6:43;
     reconsider p as non zero Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
     the carrier of Polynom-Ring F
        c= the carrier of Polynom-Ring FAdj(F,V2) by FIELD_4:10; then
     reconsider q = p as Element of the carrier of Polynom-Ring FAdj(F,V2);
     C: Polynom-Ring F is Subring of Polynom-Ring FAdj(F,V2) by FIELD_4:def 1;
     now assume q is zero;
       then q = 0_.FAdj(F,V2) by UPROOTS:def 5
             .= 0.(Polynom-Ring FAdj(F,V2)) by POLYNOM3:def 10
             .= 0.(Polynom-Ring F) by C,C0SP1:def 3
             .= 0_.(F) by POLYNOM3:def 10;
       hence contradiction;
       end; then
     reconsider q as non zero Element of
                                  the carrier of Polynom-Ring FAdj(F,V2);
     Ext_eval(q,b) = 0.E2 by B,FIELD_7:15; then
     reconsider b1 = b as FAdj(F,V2)-algebraic Element of E2 by FIELD_6:43;
     thus FAdj(F,T1) = FAdj(FAdj(F,V1),{a1}) by C1,FIELD_7:34
                    .= FAdj(FAdj(F,V2),{b1}) by C4,FIELD_10:9
                    .= FAdj(F,T2) by C3,FIELD_7:34;
     end;
   hence P[k+1];
   end;
I: for k being Nat holds P[k] from NAT_1:sch 2(A,B);
consider n being Nat such that H: card T1 = n;
thus thesis by H,AS,I;
end;
