
theorem lemma6:
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) c= the carrier of U
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;
A: now assume A1: p splits_in U; 
   now let o be object;
     assume o in Roots(E,p); then o in
     {a where a is Element of E : a is_a_root_of p,E} by FIELD_4:def 4;
     then consider a being Element of E such that
     A2: o = a & a is_a_root_of p,E;
     a in U by AS,A1,A2,lemma2;
     hence o in the carrier of U by A2;
     end;
   hence Roots(E,p) c= the carrier of U;
   end;
now assume B1: Roots(E,p) c= the carrier of U;
  now let a be Element of E;
    assume a is_a_root_of p,E;
    then a in {a where a is Element of E : a is_a_root_of p,E};
    then a in Roots(E,p) by FIELD_4:def 4;
    hence a in U by B1;
    end;
  hence p splits_in U by AS,lemma2;
  end;
hence thesis by A;
end;
