
theorem sp1:
for F being Field,
    E being FieldExtension of F,
    K being E-extending FieldExtension of F
for BE being Subset of VecSp(K,E), BF being Subset of VecSp(K,F) st BF c= BE
for l being Linear_Combination of BF holds l is Linear_Combination of BE
proof
let F be Field, E be FieldExtension of F, K be E-extending FieldExtension of F;
let BE be Subset of VecSp(K,E), BF be Subset of VecSp(K,F);
assume AS: BF c= BE;
let l be Linear_Combination of BF;
reconsider l1 = l as Linear_Combination of VecSp(K,E) by sp0;
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;
now let o be object;
  assume o in Carrier l1; then
  consider v1 being Element of VecSp(K,E) such that
  A: o = v1 & l1.v1 <> 0.E by VECTSP_6:1;
  reconsider v = v1 as Element of VecSp(K,F) by H0;
  0.E = 0.F by H1,C0SP1:def 3; then
  B: v in Carrier l by A,VECTSP_6:1;
  Carrier l c= BF by VECTSP_6:def 4;
  hence o in BE by B,A,AS;
  end;
then Carrier l1 c= BE;
hence thesis by VECTSP_6:def 4;
end;
