
theorem lift1:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
for L being F-monomorphic algebraic-closed Field,
    f being Monomorphism of F,L
ex g being Function of FAdj(F,{a}),L st g is monomorphism f-extending
proof
let F be Field, E be FieldExtension of F, a be F-algebraic Element of E;
let L be F-monomorphic algebraic-closed Field;
let f be Monomorphism of F,L;
per cases;
suppose a in F; then
  {a} c= the carrier of F by TARSKI:def 1; then
  I: the doubleLoopStr of FAdj(F,{a}) = the doubleLoopStr of F
     by FIELD_7:3,FIELD_7:def 1; then
  reconsider f1 = f as Function of FAdj(F,{a}),L;
  take f1;
  f1 is additive multiplicative unity-preserving
     proof
     now let x,y be Element of FAdj(F,{a});
       reconsider x1 = x, y1 = y as Element of F by I;
       x + y = x1 + y1 by I;
       hence f1.(x+y) = f1.x + f1.y by VECTSP_1:def 20;
       end;
     hence f1 is additive;
     now let x,y be Element of FAdj(F,{a});
       reconsider x1 = x, y1 = y as Element of F by I;
       x * y = x1 * y1 by I;
       hence f1.(x*y) = f1.x * f1.y by GROUP_6:def 6;
       end;
     hence f1 is multiplicative;
     1_FAdj(F,{a}) = 1_F by I;
     hence thesis by GROUP_1:def 13;
     end;
  hence thesis;
  end;
suppose not a in F;
  reconsider L1 = L as F-homomorphic F-monomorphic FieldExtension of L
    by FIELD_4:6;
  reconsider f1 = f as Monomorphism of F,L1;
  reconsider p = MinPoly(a,F) as Element of the carrier of Polynom-Ring F;
  reconsider q = (PolyHom f1).p as Polynomial of L;
  deg q >= 0; then
  reconsider q as non constant Polynomial of L by RATFUNC1:def 2,RING_4:def 4;
  consider b being Element of L such that
  A0: b is_a_root_of q by POLYNOM5:def 8;
  eval(q,b) = 0.L by A0,POLYNOM5:def 7; then
  B0: Ext_eval(q,b) = 0.L1 by FIELD_6:10; then
  reconsider b as L-algebraic Element of L1 by FIELD_6:43;
  H: FAdj(F,{a}) = RAdj(F,{a}) by FIELD_6:56;
  set M = { [Ext_eval(p,a),Ext_eval(q,b)] where
           p is (Polynomial of F), q is Polynomial of L : q = (PolyHom f1).p};
  now let o be object;
    assume o in M; then
    consider p being (Polynomial of F), q being Polynomial of L such that
    A: o = [Ext_eval(p,a),Ext_eval(q,b)] & q = (PolyHom f1).p;
    Ext_eval(p,a) in the set of all Ext_eval(p,a) where p is Polynomial of F;
    then B: Ext_eval(p,a) in the carrier of FAdj(F,{a}) by H,FIELD_6:45;
    thus o in [:the carrier of FAdj(F,{a}),the carrier of L:]
       by A,B,ZFMISC_1:def 2;
    end;
  then M c= [:the carrier of FAdj(F,{a}),the carrier of L:];
  then reconsider M as Relation of the carrier of FAdj(F,{a}),the carrier of L;
  now let x,y1,y2 be object;
     assume A1: [x,y1] in M & [x,y2] in M; then
     consider p1 being (Polynomial of F), q1 being Polynomial of L such that
     A2: [x,y1] = [Ext_eval(p1,a),Ext_eval(q1,b)] & q1 = (PolyHom f1).p1;
     A3: x = Ext_eval(p1,a) & y1 = Ext_eval(q1,b) by A2,XTUPLE_0:1;
     consider p2 being (Polynomial of F), q2 being Polynomial of L such that
     A4: [x,y2] = [Ext_eval(p2,a),Ext_eval(q2,b)] & q2 = (PolyHom f1).p2 by A1;
     A5: x = Ext_eval(p2,a) & y2 = Ext_eval(q2,b) by A4,XTUPLE_0:1;
     Ext_eval(p1,a) = Ext_eval(p2,a) by A2,XTUPLE_0:1,A5; then
     A6: 0.E = Ext_eval(p1,a) - Ext_eval(p2,a) by RLVECT_1:15
            .= Ext_eval(p1-p2,a) by FIELD_6:27;
     reconsider pm = p1 - p2 as Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
     A7: (PolyHom f1).p divides (PolyHom f1).pm by A6,FIELD_6:53,F814;
     A8: Ext_eval(q1-q2,b) = 0.L
         proof
         reconsider u = (PolyHom f1).p as Polynomial of L;
         consider v being Polynomial of L such that
         V: u *' v = (PolyHom f1).pm by A7,RING_4:1;
         reconsider p1a = p1, p2a = p2 as
              Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
         -p2 = -p2a by FIELD_8:1; then
         Z: p1 - p2 = p1a - p2a by POLYNOM3:def 10;
         Y: -q2 = -(PolyHom f1).p2a by A4,FIELD_8:1;
         X: q1-q2
             = (PolyHom f1).p1a + -(PolyHom f1).p2a by Y,A2,POLYNOM3:def 10
            .= (PolyHom f1).p1a + (PolyHom f1).(-p2a) by F815
            .= (PolyHom f1).pm by Z,FIELD_1:24;
         L is Subfield of L1 by FIELD_4:7; then
         W: L is Subring of L1 by FIELD_5:12;
         Ext_eval((PolyHom f1).pm,b)
              = (0.L1) * Ext_eval(v,b) by B0,V,W,ALGNUM_1:20;
         hence thesis by X;
         end;
     0.L = Ext_eval(q1,b) - Ext_eval(q2,b) by A8,FIELD_6:27;
     hence y1 = y2 by A5,A3,RLVECT_1:21;
     end; then
  reconsider g = M as Function by FUNCT_1:def 1;
    A1: for o being object st o in dom M holds o in the carrier of FAdj(F,{a});
    now let o be object;
     assume o in the carrier of FAdj(F,{a}); then
     o in the set of all Ext_eval(p,a) where p is Polynomial of F
        by H,FIELD_6:45; then
     consider p being Polynomial of F such that A: o = Ext_eval(p,a);
     ex y being object st [o,y] in g
       proof
       reconsider p1 = p as Element of the carrier of Polynom-Ring F
          by POLYNOM3:def 10;
       reconsider q = (PolyHom f1).p1 as Polynomial of L;
       take y = Ext_eval(q,b);
       thus thesis by A;
       end;
    hence o in dom g by XTUPLE_0:def 12;
     end; then
    A2: dom g = the carrier of FAdj(F,{a}) by A1,TARSKI:2;
    rng M is Subset of the carrier of L; then
  reconsider g as Function of the carrier of FAdj(F,{a}),the carrier of L
                                                             by A2,FUNCT_2:2;
  U: for p being (Polynomial of F), q being Polynomial of L
     st q = (PolyHom f1).p holds g.Ext_eval(p,a) = Ext_eval(q,b)
     proof
     let p be (Polynomial of F), q be Polynomial of L;
     assume q = (PolyHom f1).p; then
     [Ext_eval(p,a),Ext_eval(q,b)] in g;
     hence thesis by FUNCT_1:1;
     end;

  take g;
  I4: L is Subring of L1 by FIELD_4:def 1;
  C1: now let x,y be Element of FAdj(F,{a});
     x in the carrier of FAdj(F,{a}); then
     x in the set of all Ext_eval(p,a) where p is Polynomial of F
         by H,FIELD_6:45; then
     consider p being Polynomial of F such that A1: x = Ext_eval(p,a);
     y in the carrier of FAdj(F,{a}); then
     y in the set of all Ext_eval(p,a) where p is Polynomial of F
         by H,FIELD_6:45; then
     consider q being Polynomial of F such that A2: y = Ext_eval(q,a);
     reconsider pF = p, qF = q as Element of the carrier of Polynom-Ring F
                                                          by POLYNOM3:def 10;
     reconsider f1p = (PolyHom f1).pF,f1q = (PolyHom f1).qF as Polynomial of L;
     B: g.x = Ext_eval(f1p,b) & g.y = Ext_eval(f1q,b) by A1,A2,U;
     D: x + y = Ext_eval(p+q,a) by A1,A2,FIELD_8:47;
     reconsider pqF = p + q as Element of the carrier of Polynom-Ring F
                                                          by POLYNOM3:def 10;
     reconsider f1pq = (PolyHom f1).(pF+qF) as Polynomial of L;
     E: (PolyHom f1).(pF+qF)
         = (PolyHom f1).pF + (PolyHom f1).qF by FIELD_1:24
        .= f1p + f1q by POLYNOM3:def 10;
     F: p + q = pF + qF by POLYNOM3:def 10;
     thus g.(x+y) = Ext_eval(f1p+f1q,b) by D,U,E,F
                 .= g.x + g.y by B,I4,ALGNUM_1:15;
     end;
  C2: now let x,y be Element of FAdj(F,{a});
     x in the carrier of FAdj(F,{a}); then
     x in the set of all Ext_eval(p,a) where p is Polynomial of F
         by H,FIELD_6:45; then
     consider p being Polynomial of F such that A1: x = Ext_eval(p,a);
     y in the carrier of FAdj(F,{a}); then
     y in the set of all Ext_eval(p,a) where p is Polynomial of F
         by H,FIELD_6:45; then
     consider q being Polynomial of F such that A2: y = Ext_eval(q,a);
     reconsider pF = p, qF = q as Element of the carrier of Polynom-Ring F
                                                          by POLYNOM3:def 10;
     reconsider f1p = (PolyHom f1).pF,f1q = (PolyHom f1).qF as Polynomial of L;
     B: g.x = Ext_eval(f1p,b) & g.y = Ext_eval(f1q,b) by A1,A2,U;
     D: x * y = Ext_eval(p*'q,a) by A1,A2,FIELD_8:47;
     reconsider pqF = p *' q as Element of the carrier of Polynom-Ring F
                                                          by POLYNOM3:def 10;
     reconsider f1pq = (PolyHom f1).(pF*qF) as Polynomial of L;
     E: (PolyHom f1).(pF*qF)
         = (PolyHom f1).pF * (PolyHom f1).qF by FIELD_1:25
        .= f1p *' f1q by POLYNOM3:def 10;
     F: p *' q = pF * qF by POLYNOM3:def 10;
     thus g.(x*y) = Ext_eval(f1p*'f1q,b) by D,U,E,F
                 .= g.x * g.y by B,I4,ALGNUM_1:20;
     end;
  C3: g.(1_FAdj(F,{a})) = 1_L1
      proof
      reconsider g1 = (1.F)|F as
              Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
      reconsider q = (1.L)|L as
              Element of the carrier of Polynom-Ring L by POLYNOM3:def 10;
      F: f.(1_F) = 1_L by GROUP_1:def 13;
      1.FAdj(F,{a}) = Ext_eval(1_.F,a) by FIELD_8:46
                   .= Ext_eval(g1,a) by RING_4:14; then
      g.(1.FAdj(F,{a}))
          = Ext_eval((PolyHom f1).g1,b) by U
         .= Ext_eval(q,b) by F,FIELD_8:13
         .= LC((1.L)|L) by FIELD_6:28
         .= 1.L1 by FIELD_8:3;
      hence thesis;
      end;
  C: g is additive multiplicative unity-preserving by C1,C2,C3;
  now let x be Element of F;
    reconsider g1 = x|F as
              Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
    x = Ext_eval(x|F,a) by FIELD_8:48;
    hence g.x = Ext_eval((PolyHom f1).g1,b) by U
             .= Ext_eval((f.x)|L,b) by FIELD_8:13
             .= LC((f.x)|L) by FIELD_6:28
             .= f.x by FIELD_8:3;
    end;
  hence thesis by C;
  end;
end;
