
theorem 
for F being Field,
    E1 being F-finite FieldExtension of F,
    E2 being FieldExtension of F
st E1,E2 are_isomorphic_over F holds E2 is F-finite & deg(E1,F) = deg(E2,F)
proof
let F be Field, E1 be F-finite FieldExtension of F,
    E2 being FieldExtension of F;
assume E1,E2 are_isomorphic_over F; then
   consider h being Function of E1,E2 such that A: h is F-isomorphism;
   H: the carrier of VecSp(E1,F) = the carrier of E1 & 
      the carrier of VecSp(E2,F) = the carrier of E2 by FIELD_4:def 6;
   reconsider E = E2 as E1-homomorphic FieldExtension of F by A,RING_2:def 4;
   reconsider h as Homomorphism of E1,E by A;
   reconsider i = h as linear-transformation of VecSp(E1,F),VecSp(E,F)
     by A,lintrans;
   C: VecSp(E1,F) is finite-dimensional by FIELD_4:def 8;
   i is bijective by H,A; then
   D: VecSp(E2,F) is finite-dimensional & 
      dim(VecSp(E1,F)) = dim(VecSp(E2,F)) by C,VECTSP12:4;
   hence E2 is F-finite by FIELD_4:def 8;
   thus deg(E1,F) = dim(VecSp(E2,F)) by D,FIELD_4:def 7 
                 .= deg(E2,F) by D,FIELD_4:def 7;
end;
