
theorem
for F being Field,
    p being non constant Element of the carrier of Polynom-Ring F
for U being FieldExtension of F 
for E being U-extending FieldExtension of F st p splits_in E
holds p splits_in U iff Roots(E,p) = Roots(U,p)
proof
let F be Field,
    p be non constant Element of the carrier of Polynom-Ring F;
let U being FieldExtension of F;
let E being U-extending FieldExtension of F;
assume AS: p splits_in E;
now assume p splits_in U; then
   B: Roots(E,p) c= Roots(U,p) by AS,lemma3;
   Roots(U,p) c= Roots(E,p) by lemma7;
   hence Roots(E,p) = Roots(U,p) by B,XBOOLE_0:def 10;
   end;
hence thesis by AS,lemma3;
end;
