
theorem lembas2a:
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
ex p being Polynomial of F
st deg p < deg MinPoly(a,F) &
   for i being Element of NAT st i < deg MinPoly(a,F) holds p.i = l.(a|^i)
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;
not(deg MinPoly(a,F) <= 0) by RING_4:def 4; then
reconsider n = deg MinPoly(a,F) - 1 as Element of NAT by INT_1:3;
E is FAdj(F,{a})-extending by FIELD_4:7; then
B: VecSp(FAdj(F,{a}),F) is Subspace of VecSp(E,F) by FIELD_5:14;
consider l2 being Linear_Combination of VecSp(E,F) such that
K: Carrier l2 = Carrier l &
   for v being Element of VecSp(E,F) st v in Carrier l2 holds l2.v = l.v
   by B,lcsub;
consider p being Polynomial of F such that
A: deg p <= n &
   for i being Element of NAT st i <= n holds p.i = l2.(a|^i) by lembasx2;
take p;
deg p + 0 < (deg MinPoly(a,F) - 1) + 1 by A,XREAL_1:8;
hence deg p < deg MinPoly(a,F);
now let i be Element of NAT;
  assume i < deg MinPoly(a,F);
  then i < n + 1;
  then B: i <= n by NAT_1:13;
  the carrier of VecSp(FAdj(F,{a}),F) = the carrier of FAdj(F,{a})
    by FIELD_4:def 6; then
  V1: a|^i is Element of VecSp(FAdj(F,{a}),F) by lcsub1;
  V2: a|^i is Element of VecSp(E,F) by FIELD_4:def 6;
  thus p.i = l.(a|^i)
    proof
    per cases;
    suppose a|^i in Carrier l2;
      then l2.(a|^i) = l.(a|^i) by K;
      hence thesis by A,B;
      end;
    suppose C: not a|^i in Carrier l2;
      then l2.(a|^i) = 0.F by V2,VECTSP_6:2 .= l.(a|^i) by V1,C,K,VECTSP_6:2;
      hence thesis by A,B;
      end;
    end;
  end;
hence thesis;
end;
