
theorem fixr:
for F being Field,
    E being F-algebraic FieldExtension of F
for h being F-fixing Monomorphism of E holds the carrier of E c= rng h
proof
let F be Field, E being F-algebraic FieldExtension of F,
    h be F-fixing Monomorphism of E;
now let o be object;
  assume o in the carrier of E; then
  reconsider a = o as Element of E;
  set p = MinPoly(a,F), M = Roots(E,MinPoly(a,F));
  H: M = {a where a is Element of E : a is_a_root_of p,E} by FIELD_4:def 4;
  Ext_eval(p,a) = 0.E by FIELD_6:52;
  then a is_a_root_of p,E by FIELD_4:def 2;
  then a in M by H;
  then a in h.:M by fixrr;
  then consider x being object such that
  A: x in dom h & x in M & a = h.x by FUNCT_1:def 6;
  thus o in rng h by A,FUNCT_1:def 3;
  end;
hence thesis;
end;
