
theorem lintrans:
for F being Field,
    E1 being FieldExtension of F,
    E2 being E1-homomorphic FieldExtension of F
for h being Homomorphism of E1,E2 
holds h is F-fixing iff
      h is linear-transformation of VecSp(E1,F),VecSp(E2,F)
proof
let F be Field, E1 be FieldExtension of F, 
    E2 be E1-homomorphic FieldExtension of F;
let h be Homomorphism of E1,E2;
G: now assume AS: h is F-fixing;
   H: dom h = the carrier of E1 by FUNCT_2:def 1
           .= the carrier of VecSp(E1,F) by FIELD_4:def 6;
   rng h c= the carrier of E2 by RELAT_1:def 19; then
   rng h c= the carrier of VecSp(E2,F) by FIELD_4:def 6; then
   reconsider f = h as Function of VecSp(E1,F),VecSp(E2,F) by H,FUNCT_2:2;
   now let x,y be Element of VecSp(E1,F);
     reconsider a = x, b = y as Element of E1 by FIELD_4:def 6;
     H: x + y = a + b by FIELD_4:def 6;
     thus f.(x+y) = h.a + h.b by H,VECTSP_1:def 20 
                 .= f.x + f.y by FIELD_4:def 6;
     end; then
   A: f is additive; 
   now let a be Scalar of F, x be Vector of VecSp(E1,F);
     reconsider v = x as Element of E1 by FIELD_4:def 6;
     F is Subring of E1 by FIELD_4:def 1; then
     the carrier of F c= the carrier of E1 by C0SP1:def 3; then
     reconsider u1 = a as Element of E1;
     F is Subring of E2 by FIELD_4:def 1; then
     the carrier of F c= the carrier of E2 by C0SP1:def 3; then
     reconsider u2 = a as Element of E2;
     I: [u1,v] in [:the carrier of F,the carrier of E1:] by ZFMISC_1:def 2;
     H: a * x = (the multF of E1)|[:the carrier of F,the carrier of E1:].(u1,v)
                by FIELD_4:def 6
             .= u1 * v by I,FUNCT_1:49; 
     J: [u2,h.v] in [:the carrier of F,the carrier of E2:] by ZFMISC_1:def 2;
     thus f.(a*x) 
        = h.u1 * h.v by H,GROUP_6:def 6
       .= u2 * h.v by AS
       .= (the multF of E2)|[:the carrier of F,the carrier of E2:].(u2,h.v)
          by J,FUNCT_1:49
       .= a * f.x by FIELD_4:def 6;
     end;
   hence h is linear-transformation of VecSp(E1,F),VecSp(E2,F) 
     by A,MOD_2:def 2;
   end;
set V1 = VecSp(E1,F), V2 = VecSp(E2,F);
now assume h is linear-transformation of VecSp(E1,F),VecSp(E2,F); then
  reconsider h1 = h as linear-transformation of VecSp(E1,F),VecSp(E2,F);
  A: the carrier of E1 = the carrier of VecSp(E1,F) by FIELD_4:def 6; 
  reconsider 1V = 1.E1 as Element of VecSp(E1,F) by FIELD_4:def 6;
  now let a be Element of F;
    F is Subfield of E1 by FIELD_4:7; then
    B: the carrier of F c= the carrier of E1 &
       the multF of F = (the multF of E1) || the carrier of F &
       1.F = 1.E1 by EC_PF_1:def 1; 
    F is Subfield of E2 by FIELD_4:7; then
    the carrier of F c= the carrier of E2 by EC_PF_1:def 1; 
    then reconsider a2 = a as Element of E2;
    reconsider aV = a as Element of VecSp(E1,F) by A,B;
    D: [a,1.F] in [:the carrier of F,the carrier of E1:] by B,ZFMISC_1:def 2;
    E: [a,1.F] in [:the carrier of F,the carrier of F:] by ZFMISC_1:def 2;
    F: (the multF of F).(a,1.F)
        = (the multF of E1).(a,1.F) by B,E,FUNCT_1:49
       .= ((the multF of E1)|[:the carrier of F,the carrier of E1:]).(a,1.F)
          by D,FUNCT_1:49
       .= (the lmult of V1).(a,1V) by B,FIELD_4:def 6; 
    I: [a,1.E2] in [:the carrier of F,the carrier of E2:] by ZFMISC_1:def 2;
    G: h1.1V = h.(1_E1) .= 1_E2 by GROUP_1:def 13;
    J: (the lmult of V2).(a,1.E2)
         = ((the multF of E2)|[:the carrier of F,the carrier of E2:]).(a,1.E2) 
           by FIELD_4:def 6
        .= (the multF of E2).(a2,1.E2) by I,FUNCT_1:49;
    h.(a * 1.F) = h1.(a * 1V) by F
               .= a * h1.1V by MOD_2:def 2 
               .= a2 * 1.E2 by J,G;
    hence h.a = a;
    end;
  hence h is F-fixing;
  end;
hence thesis by G;
end;
