
theorem lemma7:
for F being Field,
    p being Element of the carrier of Polynom-Ring F
for E being FieldExtension of F,
    U being E-extending FieldExtension of F
holds Roots(E,p) c= Roots(U,p)
proof
let F be Field,
    p be Element of the carrier of Polynom-Ring F;
let E be FieldExtension of F, U be E-extending FieldExtension of F;
E is Subfield of U by FIELD_4:7; then
H: the carrier of E c= the carrier of U & 0.E = 0.U by EC_PF_1:def 1;
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
  A: o = a & a is_a_root_of p,E;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
     by FIELD_4:10; then
  reconsider p1 = p as Element of the carrier of (Polynom-Ring E);
  reconsider U1 = U as FieldExtension of E;
  reconsider b = a as Element of U by H;
  Ext_eval(p,b) = Ext_eval(p,a) by FIELD_7:14
               .= 0.U by H,A,FIELD_4:def 2;
  then b is_a_root_of p,U by FIELD_4:def 2; then
  a in {a where a is Element of U : a is_a_root_of p,U};
  hence o in Roots(U,p) by A,FIELD_4:def 4;
  end;
hence thesis;
end;
