
theorem lemNor2b:
for F being Field,
    E being FieldExtension of F,
    K being F-extending FieldExtension of E
for h being F-fixing Monomorphism of E,K
for p being non constant Element of the carrier of Polynom-Ring F
st p splits_in E holds h .: Roots(E,p) c= the carrier of E
proof
let F be Field, E be FieldExtension of F,
    K be F-extending FieldExtension of E;
let h be F-fixing Monomorphism of E,K;
let p be non constant Element of the carrier of Polynom-Ring F;
assume AS: p splits_in E;
H: Roots(E,p) = {a where a is Element of E : a is_a_root_of p,E} &
   Roots(K,p) = {a where a is Element of K : a is_a_root_of p,K}
   by FIELD_4:def 4;
now let o be object;
  assume o in h.:Roots(E,p); then
  consider x being object such that
  B: x in dom h & x in Roots(E,p) & o = h.x by FUNCT_1:def 6;
  consider a being Element of E such that
  C: a = x & a is_a_root_of p,E by B,H;
  Ext_eval(p,a) = 0.E by C,FIELD_4:def 2; then
  0.K = h.(Ext_eval(p,a)) by RING_2:6 .= Ext_eval(p,h.a) by fixeval; then
  h.a is_a_root_of p,K by FIELD_4:def 2; then
  h.a in Roots(K,p) by H; then
  h.a in Roots(E,p) by AS,lemNor2bb;
  hence o in the carrier of E by C,B;
  end;
hence thesis;
end;
