
theorem uu0:
for F1 being Field,
    F2 being F1-isomorphic F1-homomorphic Field
for h being Isomorphism of F1,F2
for p,q being Element of the carrier of Polynom-Ring F1
holds p divides q iff (PolyHom h).p divides (PolyHom h).q
proof
let F1 be Field, F2 be F1-isomorphic F1-homomorphic Field;
let h be Isomorphism of F1,F2;
let p,q be Element of the carrier of Polynom-Ring F1;
A: now assume AS: p divides q;
   reconsider pp = p, qq = q as Polynomial of F1;
   consider rr being Polynomial of F1 such that
   B: pp *' rr = qq by AS,RING_4:1;
   reconsider r = rr as Element of the carrier of Polynom-Ring F1
     by POLYNOM3:def 10;
   p * r = q by B,POLYNOM3:def 10; then
   C: (PolyHom h).q = (PolyHom h).p * (PolyHom h).r by FIELD_1:25;
   reconsider pe = (PolyHom h).p, qe = (PolyHom h).q, re = (PolyHom h).r
     as Polynomial of F2;
   qe = pe *' re by C,POLYNOM3:def 10;
   hence (PolyHom h).p divides (PolyHom h).q by RING_4:1;
   end;
reconsider f = (PolyHom h)" as
        Isomorphism of Polynom-Ring F2,Polynom-Ring F1 by RING_3:73;
now assume AS: (PolyHom h).p divides (PolyHom h).q;
   reconsider pp = (PolyHom h).p, qq = (PolyHom h).q as Polynomial of F2;
   consider rr being Polynomial of F2 such that
   B: pp *' rr = qq by AS,RING_4:1;
   reconsider r = rr as Element of the carrier of Polynom-Ring F2
     by POLYNOM3:def 10;
   (PolyHom h).p * r = (PolyHom h).q by B,POLYNOM3:def 10; then
   C: f.((PolyHom h).p) * f.r = f.((PolyHom h).q) by GROUP_6:def 6;
   D: f.((PolyHom h).p) = p & f.((PolyHom h).q) = q by FUNCT_2:26;
   reconsider pe = p, qe = q, re = f.r as Polynomial of F1;
   qe = pe *' re by C,D,POLYNOM3:def 10;
   hence p divides q by RING_4:1;
   end;
hence thesis by A;
end;
