
theorem T1:
for F being Field,
    E being F-finite FieldExtension of F,
    K being E-finite F-extending FieldExtension of E
for BE being Basis of VecSp(E,F),
    BK being Basis of VecSp(K,E)
for l being Linear_Combination of Base(BE,BK)
st Sum(l) = 0.VecSp(K qua FieldExtension of F,F) holds Carrier(l) = {}
proof
let F be Field, E be F-finite FieldExtension of F,
    K be E-finite F-extending FieldExtension of E;
let BE be Basis of VecSp(E,F), BK be Basis of VecSp(K,E);
let l1 be Linear_Combination of Base(BE,BK);
set l2 = down l1;
assume AS: Sum l1 = 0.VecSp((K qua FieldExtension of F),F);
B: Sum l2 = Sum l1 by TSum
         .= 0.K by AS,FIELD_4:def 6
         .= 0.VecSp(K,E) by FIELD_4:def 6;
C: Carrier(l2) = {} by B,VECTSP_7:def 1;
D:now let x be Element of K;
  assume x in Base(BE,BK); then
  consider a,b being Element of K such that
  B1: x = a * b & a in BE & b in BK;
  reconsider bv = b as Element of VecSp(K,E) by FIELD_4:def 6;
  0.VecSp(E,F) = 0.E by FIELD_4:def 6
              .= l2.bv by C,VECTSP_6:2
              .= Sum down(l1,b) by B1,down2; then
  B3: Carrier down(l1,b) = {} by VECTSP_7:def 1;
  reconsider av = a as Element of VecSp(E,F) by B1;
  0.F = down(l1,b).av by B3,VECTSP_6:2 .= l1.(a*b) by B1,down1;
  hence not x in Carrier l1 by B1,VECTSP_6:2;
  end;
now let o be object;
  assume C1: o in Carrier l1; then
  reconsider x = o as Element of K by FIELD_4:def 6;
  Carrier l1 c= Base(BE,BK) by VECTSP_6:def 4; then
  x in Base(BE,BK) by C1;
  hence contradiction by D,C1;
  end;
hence thesis by XBOOLE_0:def 1;
end;
