
theorem
for F1 being Field,
    p being non constant Element of the carrier of Polynom-Ring F1
for F2 being F1-isomorphic Field
for h being Isomorphism of F1,F2
holds p splits_in F1 iff (PolyHom h).p splits_in F2
proof
let F1 be Field,
    p be non constant Element of the carrier of Polynom-Ring F1;
let F2 be F1-isomorphic Field; let h be Isomorphism of F1,F2;
now assume AS: (PolyHom h).p splits_in F2;
  h" is Isomorphism of F2,F1 by RING_3:73; then
  reconsider F1a = F1 as F2-isomorphic F2-homomorphic Field
     by RING_3:def 4,RING_2:def 4;
  reconsider g = h" as Isomorphism of F2,F1a by RING_3:73;
  now let i be Element of NAT;
    thus ((PolyHom g).((PolyHom h).p)).i
       = g.(((PolyHom h).p).i) by FIELD_1:def 2
      .= g.(h.(p.i)) by FIELD_1:def 2
      .= p.i by FUNCT_2:26;
    end;
  then (PolyHom g).((PolyHom h).p) = p;
  hence p splits_in F1 by AS,lemma6a;
  end;
hence thesis by lemma6a;
end;
