
theorem lembas2bb:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E
for l being Linear_Combination of Base a,
    p being non zero Polynomial of F
st l.(a|^(deg p)) = LC p & Carrier l = {a|^(deg p)}
holds Sum l = Ext_eval(LM p,a)
proof
let F be Field, E be FieldExtension of F;
let a be F-algebraic Element of E;
let l be Linear_Combination of Base a, p be non zero Polynomial of F;
F is Subring of E by FIELD_4:def 1; then
H2: the carrier of F c= the carrier of E by C0SP1:def 3;
H3: {a} is Subset of FAdj(F,{a}) by FAt;
    a in {a} by TARSKI:def 1; then
    reconsider a1 = a as Element of FAdj(F,{a}) by H3;
H7: FAdj(F,{a}) is Subring of E by FIELD_5:12;
assume A: l.(a|^(deg p)) = LC p & Carrier l = {a|^(deg p)};
reconsider LCpE = LC p as Element of E by H2;
F is Subfield of FAdj(F,{a}) by FAsub; then
the carrier of F c= the carrier of FAdj(F,{a}) by EC_PF_1:def 1; then
reconsider LCp = LC p as Element of FAdj(F,{a});
reconsider degp = deg p as Nat;
H8: a|^(degp) = a1|^(degp) by H7,pr5; then
reconsider adegp = a|^(deg p) as Element of FAdj(F,{a});
reconsider v = a|^(deg p) as Element of VecSp(FAdj(F,{a}),F)
by H8,FIELD_4:def 6;
B: E is FieldExtension of FAdj(F,{a}) by FIELD_4:7;
Sum l = l.v * v by A,VECTSP_6:20
     .= LCp * adegp by A,Lm12a
     .= LCpE * (a|^(deg p)) by B,Lm12b
     .= Ext_eval(LM p,a) by exevalLM;
hence thesis;
end;
