
theorem lembas:
for F being Field,
    E being (Polynom-Ring F)-homomorphic FieldExtension of F
for a being F-algebraic Element of E holds
Base a is Basis of VecSp(FAdj(F,{a}),F)
proof
let F be Field, E be (Polynom-Ring F)-homomorphic FieldExtension of F;
let a be F-algebraic Element of E;
set V = VecSp(FAdj(F,{a}),F), ma = MinPoly(a,F);
H0: F is Subring of E by FIELD_4:def 1;
A: now let l be Linear_Combination of Base a;
   assume A1: Sum(l) = 0.V;
   0.V = 0.(FAdj(F,{a})) by FIELD_4:def 6
      .= 0.E by dFA .= 0.F by H0,C0SP1:def 3;
   then l = ZeroLC V by A1,lembas2;
   hence Carrier(l) = {} by VECTSP_6:def 3;
   end;
E: now let o be object;
   assume o in the carrier of the ModuleStr of V; then
   o in the carrier of FAdj(F,{a}) by FIELD_4:def 6; then
   o in the carrier of RAdj(F,{a}) by ch1; then
   o in the set of all Ext_eval(p,a) where p is Polynomial of F by lemphi5;
   then consider p being Polynomial of F such that A1: o = Ext_eval(p,a);
   o in Lin(Base a) by A1,lembas1;
   hence o in the carrier of Lin Base a;
   end;
now let o be object;
  assume G: o in the carrier of Lin Base a;
  the carrier of Lin Base a c= the carrier of V by VECTSP_4:def 2;
  hence o in the carrier of V by G;
  end;
then Lin Base a = the ModuleStr of V by VECTSP_4:31,E,TARSKI:2;
hence thesis by A,VECTSP_7:def 3,VECTSP_7:def 1;
end;
