
theorem F814:
for F1 being Field,
    F2 being F1-monomorphic F1-homomorphic Field
for h being Monomorphism of F1,F2
for p,q being Element of the carrier of Polynom-Ring F1
holds p divides q implies (PolyHom h).p divides (PolyHom h).q
proof
let F1 be Field, F2 be F1-monomorphic F1-homomorphic Field;
let h be Monomorphism of F1,F2;
let p,q be Element of the carrier of Polynom-Ring F1;
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;
