
theorem sp0:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for l being Linear_Combination of VecSp(K,F)
holds l is Linear_Combination of VecSp(K,E)
proof
let F be Field, E be FieldExtension of F, K be E-extending FieldExtension of F;
let l be Linear_Combination of VecSp(K,F);
H0: the carrier of VecSp(K,F) = the carrier of K by FIELD_4:def 6
                             .= the carrier of VecSp(K,E) by FIELD_4:def 6;
H1: F is Subring of E & E is Subring of K by FIELD_4:def 1; then
H2: the carrier of F c= the carrier of E &
    the carrier of E c= the carrier of K by C0SP1:def 3;
rng l c= the carrier of E by H2; then
l is Function of the carrier of VecSp(K,E),the carrier of E by H0,FUNCT_2:6;
then A: l in Funcs(the carrier of VecSp(K,E),the carrier of E) by FUNCT_2:8;
consider T being finite Subset of VecSp(K,F) such that
B: for v being Element of VecSp(K,F) st not v in T holds l.v = 0.F
   by VECTSP_6:def 1;
reconsider T1 = T as finite Subset of VecSp(K,E) by H0;
now let v1 be Element of VecSp(K,E);
  assume C: not v1 in T1;
  reconsider v = v1 as Element of  VecSp(K,F) by H0;
  l.v = 0.F by B,C;
  hence l.v1 = 0.E by H1,C0SP1:def 3;
  end;
hence thesis by A,VECTSP_6:def 1;
end;
