
theorem simpA:
for F being 0-characteristic Field,
    E being FieldExtension of F
for a,b being F-algebraic Element of E
ex x being Element of F st FAdj(F,{a,b}) = FAdj(F,{a+@(x,E)*b})
proof
let F be 0-characteristic Field, E be FieldExtension of F,
    a,b be F-algebraic Element of E;
  set K = FAdj(F,{a,b}), ma = MinPoly(a,F), mb = MinPoly(b,F);
  {a,b} is Subset of K & a in {a,b} & b in {a,b} by FIELD_6:35,TARSKI:def 2;
  then reconsider aK = a, bK = b as Element of K;
  consider Z being FieldExtension of E such that
  V: Z is algebraic-closed by FIELD_12:28;
  S: E is Subfield of Z by FIELD_4:7; then
  U: K is Subfield of Z by EC_PF_1:5; then
  reconsider Z as K-extending FieldExtension of F by FIELD_4:7;
  set Rma = Roots(Z,ma), Rmb = Roots(Z,mb) \ {b};
  ex x being Element of F st
     for c,d being Element of Z st c in Rma & d in Rmb
     holds @(aK,Z) + @(x,Z) * @(bK,Z) <> c + @(x,Z) * d
    proof
    per cases;
    suppose W: Rma = {} or Rmb = {};
      take 1.F;
      thus thesis by W;
      end;
    suppose W: Rma <> {} & Rmb <> {};
    set M = { (c - @(aK,Z)) * (@(bK,Z) - d)"
                          where c,d is Element of Z : c in Rma & d in Rmb};
      M <> {}
      proof
      set c = the Element of Rma, d = the Element of Rmb;
      c in Rma & d in Rmb by W; then
      reconsider c,d as Element of Z;
      (c - @(aK,Z)) * (@(bK,Z) - d)" in M by W;
      hence thesis;
      end;
    then reconsider M as non empty set;
    F: M is finite
       proof
       defpred P[object,object] means
          ex c,d being Element of Z st c in Rma & d in Rmb &
          $1 = [c,d] & $2 = (c - @(aK,Z)) * (@(bK,Z) - d)";
       F1: now let x be object;
           assume x in [:Rma,Rmb:]; then
           consider c,d being object such that
           F2: c in Rma & d in Rmb & x = [c,d] by ZFMISC_1:def 2;
           reconsider c,d as Element of Z by F2;
           (c - @(aK,Z)) * (@(bK,Z) - d)" in M by F2;
           hence ex y being object st y in M & P[x,y] by F2;
           end;
       consider f being Function of [:Rma,Rmb:],M such that
       F2: for x being object st x in [:Rma,Rmb:] holds P[x,f.x]
           from FUNCT_2:sch 1(F1);
       M c= rng f
           proof
           now let o be object;
             assume o in M; then
             consider c,d being Element of Z such that
             F5: o = (c - @(aK,Z)) * (@(bK,Z) - d)" & c in Rma & d in Rmb;
             F6: [c,d] in [:Rma,Rmb:] by F5,ZFMISC_1:def 2; then
             F7: [c,d] in dom f by FUNCT_2:def 1;
             P[[c,d],f.[c,d]] by F6,F2; then
             consider c1,d1 being Element of Z such that
             F8: [c1,d1] = [c,d] & f.[c1,d1] = (c1-@(aK,Z)) * (@(bK,Z)-d1)";
             c1 = c & d1 = d by F8, XTUPLE_0:1;
             hence o in rng f by F8,F5,F7,FUNCT_1:def 3;
             end;
           hence thesis;
           end;
       hence thesis;
       end;
    now assume G: for x being Element of F ex c,d being Element of Z
        st c in Rma & d in Rmb & @(aK,Z) + @(x,Z) * @(bK,Z) = c + @(x,Z) * d;
     now let o be object;
       assume o in the carrier of F; then
       reconsider x = o as Element of F;
       consider c,d being Element of Z such that
       G1: c in Rma & d in Rmb &
           @(aK,Z) + @(x,Z) * @(bK,Z) = c + @(x,Z) * d by G;
       G3: now assume @(bK,Z) - d = 0.Z;
           then @(bK,Z) = 0.Z + d by VECTSP_2:2;
           then d = b by FIELD_7:def 4;
           then not b in {b} by G1,XBOOLE_0:def 5;
           hence contradiction by TARSKI:def 1;
           end;
       @(x,Z) * @(bK,Z) = (c + @(x,Z) * d) - @(aK,Z) by G1,VECTSP_2:2
                       .= (c + -@(aK,Z)) + @(x,Z) * d by RLVECT_1:def 3; then
       c - @(aK,Z) = @(x,Z) * @(bK,Z) - @(x,Z) * d by VECTSP_2:2
                  .= @(x,Z) * (@(bK,Z) - d) by VECTSP_1:11; then
       (c - @(aK,Z)) * (@(bK,Z) - d)"
           = @(x,Z) * ((@(bK,Z) - d) * (@(bK,Z) - d)") by GROUP_1:def 3
          .= @(x,Z) * 1.Z by G3,VECTSP_2:def 2
          .= x by FIELD_7:def 4;
       hence o in M by G1;
       end;
     then the carrier of F c= M;
     hence F is finite by F;
     end;
    hence thesis;
    end;
    end; then
  consider x being Element of F such that
  A: for c,d being Element of Z st c in Rma & d in Rmb
     holds @(aK,Z) + @(x,Z) * @(bK,Z) <> c + @(x,Z) * d;
  set lZ = @(aK,Z) + @(x,Z) * @(bK,Z);
  set G = FAdj(F,{lZ});
  reconsider G as FieldExtension of F;
  B: G is Subfield of K
     proof
     H: F is Subfield of K & F is Subfield of FAdj(F,{lZ})
         by FIELD_4:7; then
     the carrier of F c= the carrier of K by EC_PF_1:def 1; then
     reconsider x1 = x as Element of K;
     I: K is Subring of Z & @(x,Z) = x & @(bK,Z) = bK & @(aK,Z) = aK
        by U,FIELD_5:12,FIELD_7:def 4; then
     x1 * bK = @(x,Z) * @(bK,Z) by FIELD_6:16; then
     aK + x1 * bK = lZ by I,FIELD_6:15; then
     lZ in K; then
     {lZ} c= the carrier of K by TARSKI:def 1;
     hence thesis by H,U,FIELD_6:37;
     end; then
  reconsider K as G-extending FieldExtension of F by FIELD_4:7;
  reconsider Z as K-extending G-extending FieldExtension of F by FIELD_4:7;
  reconsider aK,bK as Element of K;
  {lZ} is Subset of G & lZ in {lZ} by FIELD_6:35,TARSKI:def 1; then
  reconsider lZ as G-membered Element of Z by FIELD_7:def 5;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring G
    by FIELD_4:10; then
  MinPoly(a,F) in the carrier of Polynom-Ring G &
  MinPoly(b,F) in the carrier of Polynom-Ring G; then
  reconsider maG = MinPoly(a,F), mbG = MinPoly(b,F)
                                     as Polynomial of G by POLYNOM3:def 10;
  maG is Element of the carrier of Polynom-Ring G &
  mbG is Element of the carrier of Polynom-Ring G by POLYNOM3:def 10; then
  deg maG = deg MinPoly(a,F) & deg mbG = deg MinPoly(b,F) by FIELD_4:20;
  then reconsider maG = MinPoly(a,F), mbG = MinPoly(b,F)
            as non constant Polynomial of G by RING_4:def 4,RATFUNC1:def 2;
  set g = <%@(G,lZ),-@(x,G)%>; set h = Subst(maG,g);
  reconsider mbZ = mbG, hZ = h as Polynomial of Z by FIELD_4:8;
  E: hZ gcd mbZ = X- @(bK,Z)
     proof
     E00: deg MinPoly(b,F) > 0 & LC MinPoly(b,F) = 1.F
          by RING_4:def 4,RATFUNC1:def 7;
     H: mbZ is Element of the carrier of Polynom-Ring Z &
        F is Subfield of Z by FIELD_4:7,POLYNOM3:def 10;
     E01: deg mbZ = deg MinPoly(b,F) by H,FIELD_4:20;
          LC mbZ = LC MinPoly(b,F) by FIELD_8:5
                .= 1.Z by E00,H,EC_PF_1:def 1; then
     mbZ is non constant monic
             by RING_4:def 4,E01,RATFUNC1:def 2,RATFUNC1:def 7; then
     reconsider mbZ as Ppoly of Z by V,RING_5:70;
     now let a be Element of Z;
        H: mbZ is Element of the carrier of Polynom-Ring Z
           by POLYNOM3:def 10;
        assume a is_a_root_of mbZ; then
        0.Z = eval(mbZ,a) by POLYNOM5:def 7
           .= Ext_eval(MinPoly(b,F),a) by H,FIELD_4:26; then
        multiplicity(MinPoly(b,F),a) = 1 by mpa1,FIELD_4:def 2;
        hence multiplicity(mbZ,a) = 1 by defM;
        end; then
     reconsider mbZ1 = mbZ as Ppoly of Z,(Roots mbZ) by ZZ1;
     mbZ is Element of the carrier of Polynom-Ring Z by POLYNOM3:def 10; then
     E8: Roots mbZ = Roots(Z,mb) by FIELD_7:13;
         E9: Roots(Z,mb) = {a where a is Element of Z : a is_a_root_of mb,Z}
             by FIELD_4:def 4;
         b = @(bK,Z) by FIELD_7:def 4; then
         Ext_eval(mb,@(bK,Z)) = Ext_eval(mb,b) by FIELD_6:11
                             .= 0.E by FIELD_6:52
                             .= 0.Z by S,EC_PF_1:def 1; then
         @(bK,Z) is_a_root_of mb,Z by FIELD_4:def 2; then
     E4: @(bK,Z) in Roots mbZ by E8,E9;
         H1: @(@(G,lZ),Z) = lZ & G is Subring of Z by FIELD_5:12,FIELD_7:def 4;
         HH: @(x,G) = x by FIELD_7:def 4 .= @(x,Z) by FIELD_7:def 4;
         H2: @(-@(x,G),Z) = -@(x,G) by FIELD_7:def 4
                         .= -@(x,Z) by H1,HH,FIELD_6:17;
         H3: @(aK,Z) + @(x,Z) * @(bK,Z) + (-@(x,Z)) * @(bK,Z)
              = @(aK,Z) + @(x,Z) * @(bK,Z) + -(@(x,Z) * @(bK,Z)) by VECTSP_1:9
             .= @(aK,Z) + (@(x,Z) * @(bK,Z) + -(@(x,Z) * @(bK,Z)))
                by RLVECT_1:def 3
             .= @(aK,Z) + 0.Z by RLVECT_1:5
             .= @(aK,Z);
         H5: @(aK,Z) = a by FIELD_7:def 4;
         H4: ma is Element of the carrier of Polynom-Ring F &
         maG is Element of the carrier of Polynom-Ring G &
         hZ is Element of the carrier of Polynom-Ring Z &
         h is Element of the carrier of Polynom-Ring G by POLYNOM3:def 10; then
         eval(hZ,@(bK,Z))
                = Ext_eval(h,@(bK,Z)) by FIELD_4:26
               .= Ext_eval(maG,Ext_eval(g,@(bK,Z))) by extevalsubst
               .= Ext_eval(maG,lZ + (-@(x,Z)) * @(bK,Z)) by H1,H2,exteval2
               .= Ext_eval(ma,@(aK,Z)) by H3,H4,FIELD_8:6
               .= Ext_eval(ma,a) by H5,FIELD_6:11
               .= 0.E by FIELD_6:52
               .= 0.Z by S,EC_PF_1:def 1; then
         @(bK,Z) is_a_root_of hZ by POLYNOM5:def 7; then
     @(bK,Z) in Roots hZ by POLYNOM5:def 10; then
     E5: @(bK,Z) in (Roots mbZ) /\ (Roots hZ) by E4;
     E7: now let d be Element of Z;
         assume E10: d in Rmb;
         E9: @(aK,Z) + @(x,Z) * @(bK,Z) + (-@(x,Z)) * d
              = @(aK,Z) + @(x,Z) * @(bK,Z) + -(@(x,Z) * d) by VECTSP_1:9
             .= @(aK,Z) + (@(x,Z) * @(bK,Z) + -(@(x,Z) * d))
                by RLVECT_1:def 3;
         reconsider maG as Element of the carrier of Polynom-Ring G
            by POLYNOM3:def 10;
         now assume d in Roots hZ; then
           d is_a_root_of hZ by POLYNOM5:def 10; then
           0.Z = eval(hZ,d) by POLYNOM5:def 7
              .= Ext_eval(h,d) by H4,FIELD_4:26
              .= Ext_eval(maG,Ext_eval(g,d)) by extevalsubst
              .= Ext_eval(maG,lZ + (-@(x,Z)) * d) by H1,H2,exteval2; then
           R1: @(aK,Z) + (@(x,Z)*@(bK,Z) + -(@(x,Z)*d)) is_a_root_of maG,Z
               by E9,FIELD_4:def 2;
           Roots(Z,maG) = {a where a is Element of Z : a is_a_root_of maG,Z}
              by FIELD_4:def 4; then
           @(aK,Z) + (@(x,Z)*@(bK,Z) + -(@(x,Z)*d)) in Roots(Z,maG) by R1; then
           R3: @(aK,Z) + (@(x,Z)*@(bK,Z) + -(@(x,Z)*d)) in Rma by FIELD_8:7;
           @(aK,Z) + (@(x,Z) * @(bK,Z) + -(@(x,Z) * d)) + @(x,Z) * d
              = (@(aK,Z) + @(x,Z) * @(bK,Z)) + -(@(x,Z) * d) + @(x,Z) * d
                by RLVECT_1:def 3
             .= @(aK,Z) + @(x,Z) * @(bK,Z) + (-(@(x,Z) * d) + @(x,Z) * d)
                by RLVECT_1:def 3
             .= @(aK,Z) + @(x,Z) * @(bK,Z) + 0.Z by RLVECT_1:5;
           hence contradiction by E10,R3,A;
           end;
         hence not d in Roots hZ;
         end;
     (Roots mbZ) /\ (Roots hZ) = {@(bK,Z)}
       proof
       Z: for o being object st o in {@(bK,Z)}
          holds o in (Roots mbZ) /\ (Roots hZ) by E5,TARSKI:def 1;
       now let o be object;
          assume Z0: o in (Roots mbZ) /\ (Roots hZ); then
          reconsider z = o as Element of Z;
          Z1: z in Roots(Z,mb) & (z in Roots hZ) by Z0,E8,XBOOLE_0:def 4;
          then not z in Rmb by E7; then
          z in {b} by Z1,XBOOLE_0:def 5; then
          z = b by TARSKI:def 1 .= @(bK,Z) by FIELD_7:def 4;
          hence o in {@(bK,Z)} by TARSKI:def 1;
          end;
       hence thesis by Z,TARSKI:2;
       end; then
     E6: mbZ1 gcd hZ is Ppoly of Z,{@(bK,Z)} by simpAgcd;
     set gZ = mbZ1 gcd hZ;
     deg gZ = card {@(bK,Z)} by E6,RING_5:60 .= 1 by CARD_2:42; then
     consider x,z being Element of Z such that
     E7: x <> 0.Z & gZ = x * rpoly(1,z) by HURWITZ:28;
     E8: 1.Z = LC gZ by RATFUNC1:def 7 .= x * LC rpoly(1,z) by E7,RING_5:5
            .= x * 1.Z by RATFUNC1:def 7;
     E9: Roots gZ = {@(bK,Z)} by E6,RING_5:63;
     eval(x*rpoly(1,z),z) = z-z by E8,HURWITZ:29 .= 0.Z by RLVECT_1:15; then
     z is_a_root_of x * rpoly(1,z) by POLYNOM5:def 7; then
     z in Roots gZ by E7,POLYNOM5:def 10; then
     z = @(bK,Z) by E9,TARSKI:def 1;
     hence thesis by E7,E8,FIELD_9:def 2;
     end;
  F: b in G
     proof
     hZ is Element of the carrier of Polynom-Ring Z &
     mbZ is Element of the carrier of Polynom-Ring Z &
     h is Element of the carrier of Polynom-Ring G &
     mbG is Element of the carrier of Polynom-Ring G by POLYNOM3:def 10; then
     hZ gcd mbZ = h gcd mbG by lemgcd; then
     reconsider v = X- @(bK,Z) as Polynomial of G by E;
     G1: @(bK,Z) = bK & K is Subring of Z & K is Subring of E
         by FIELD_4:def 1,FIELD_7:def 4,FIELD_5:12;
         G is Subfield of K by FIELD_4:7; then
         G is Subfield of E by EC_PF_1:5; then
     G2: G is Subring of E by FIELD_5:12;
     v.0 = rpoly(1,@(bK,Z)).0 by FIELD_9:def 2
        .= -power(Z).(@(bK,Z),1+0) by HURWITZ:25
        .= -(power(Z).(@(bK,Z),0) * @(bK,Z)) by GROUP_1:def 7
        .= -(1_Z * @(bK,Z)) by GROUP_1:def 7
        .= -bK by G1,FIELD_6:17
        .= -b by G1,FIELD_6:17; then
     reconsider u = -b as Element of G;
     -u = --b by G2,FIELD_6:17;
     hence thesis;
     end;
  G: a in G
     proof
     {lZ} is Subset of G & lZ in {lZ} by TARSKI:def 1,FIELD_6:35; then
     reconsider bG = b, lG = lZ as Element of G by F;
     H1: G is Subring of Z by FIELD_5:12;
     H2: @(bK,Z) = bG by FIELD_7:def 4;
     @(x,Z) = x by FIELD_7:def 4 .= @(x,G) by FIELD_7:def 4; then
     @(x,Z) * @(bK,Z) = @(x,G) * bG by H1,H2,FIELD_6:16; then
     -(@(x,Z) * @(bK,Z)) = -(@(x,G) * bG) by H1,FIELD_6:17; then
     lG - @(x,G) * bG
        = (@(aK,Z) + @(x,Z) * @(bK,Z)) + -(@(x,Z) * @(bK,Z)) by H1,FIELD_6:15
       .= @(aK,Z) + (@(x,Z) * @(bK,Z) + -(@(x,Z) * @(bK,Z))) by RLVECT_1:def 3
       .= @(aK,Z) + 0.Z by RLVECT_1:5
       .= a by FIELD_7:def 4;
     hence thesis;
     end;
     G is Subfield of K by FIELD_4:7; then
     {a,b} c= the carrier of G & G is Subfield of E & F is Subfield of G
     by F,G,TARSKI:def 2,FIELD_4:7,EC_PF_1:5; then
  C: FAdj(F,{a,b}) == G by B,FIELD_6:37;
     a + @(x,E) * b = @(aK,Z) + @(x,Z) * @(bK,Z)
     proof
     E is Subfield of Z by FIELD_4:7; then
     D1: E is Subring of Z by FIELD_5:12;
     D2: @(x,E) = x by FIELD_7:def 4 .= @(x,Z) by FIELD_7:def 4;
     D3: @(bK,Z) = b & @(aK,Z) = a by FIELD_7:def 4;
     @(x,E) * b = @(x,Z) * @(bK,Z) by D1,D2,D3,FIELD_6:16;
     hence thesis by D3,D1,FIELD_6:15;
     end; then
  FAdj(F,{a,b}) = FAdj(F,{a + @(x,E) * b}) by C,FIELD_13:19;
  hence thesis;
end;
