
theorem lembasx1a:
for F being Field,
    E being FieldExtension of F
for a being Element of E
for n being Element of NAT
for l being Linear_Combination of VecSp(E,F),
    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 Element of E; let n be Element of NAT;
let l be Linear_Combination of VecSp(E,F); let 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;
assume A: l.(a|^(deg p)) = LC p & Carrier l = {a|^(deg p)};
reconsider LCp = LC p as Element of E by H2;
reconsider adegp = a|^(deg p) as Element of E;
reconsider v = a|^(deg p) as Element of VecSp(E,F) by FIELD_4:def 6;
thus Sum l = l.v * v by A,VECTSP_6:20
          .= LCp * adegp by A,Lm12a
          .= Ext_eval(LM p,a) by exevalLM;
end;
