
theorem BasKEF:
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)
holds Base(BE,BK) is Basis of VecSp(K qua FieldExtension of F,F)
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);
A: for l being Linear_Combination of Base(BE,BK)
st Sum(l) = 0.VecSp((K qua FieldExtension of F),F) holds Carrier(l) = {} by T1;
Lin Base(BE,BK) = the ModuleStr of VecSp((K qua FieldExtension of F),F) by T2;
hence thesis by A,VECTSP_7:def 1,VECTSP_7:def 3;
end;
