
theorem lemma3:
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= 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;
H: U is Subfield of E by FIELD_4:7;
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; then
     reconsider b = a as Element of U;
     Ext_eval(p,b) = Ext_eval(p,a) by FIELD_7:14
                  .= 0.E by A2,FIELD_4:def 2
                  .= 0.U by H,EC_PF_1:def 1; then
     b is_a_root_of p,U by FIELD_4:def 2; then
     b in {a where a is Element of U : a is_a_root_of p,U};
     hence o in Roots(U,p) by A2,FIELD_4:def 4;
     end;
   hence Roots(E,p) c= Roots(U,p);
   end;
now assume Roots(E,p) c= Roots(U,p);
  then Roots(E,p) c= the carrier of U by XBOOLE_1:1;
  hence p splits_in U by AS,lemma6;
  end;
hence thesis by A;
end;
