
theorem lembas2:
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
holds Sum(l) = 0.F implies l = ZeroLC VecSp(FAdj(F,{a}),F)
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;
assume AS: Sum(l) = 0.F;
set V = VecSp(FAdj(F,{a}),F), ma = MinPoly(a,F);
H: F is Subring of E by FIELD_4:def 1;
consider p being Polynomial of F such that
A2: deg p < deg MinPoly(a,F) &
    for i being Element of NAT st i < deg ma holds p.i = l.(a|^i) by lembas2a;
now assume A3: Carrier l <> {};
  set x = the Element of Carrier l;
  consider v being Element of V such that
  A5: x = v & l.v <> 0.F by A3,VECTSP_6:1;
  Carrier l c= Base a by VECTSP_6:def 4; then
  v in Base a by A3,A5; then
  consider i being Element of NAT such that A6: v=a|^i & i < deg MinPoly(a,F);
  p.i <> 0.F by A2,A6,A5; then
  p <> 0_.(F);
  then reconsider p as non zero Polynomial of F by UPROOTS:def 5;
  reconsider pp = p as non zero Element of the carrier of Polynom-Ring F
      by POLYNOM3:def 10;
  Ext_eval(pp,a) = 0.F by A2,AS,lembas2b .= 0.E by H,C0SP1:def 3;
  hence contradiction by mpol4,A2,RING_5:13;
  end;
hence thesis by VECTSP_6:def 3;
end;
