
theorem T2:
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 Lin Base(BE,BK) = the ModuleStr 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);
set V = VecSp((K qua FieldExtension of F),F);
H1: the carrier of V = the carrier of K by FIELD_4:def 6;
H2: Lin BE = the ModuleStr of VecSp(E,F) &
    Lin BK = the ModuleStr of VecSp(K,E) by VECTSP_7:def 3;
A: now let o be object;
   assume o in the carrier of V;
   then o in Lin BK by H1,H2,FIELD_4:def 6;
   then consider l1 being Linear_Combination of BK such that
   A0: o = Sum l1 by VECTSP_7:7;
   set l = lift(l1,BE);
   down l = l1 by Tlift2;
   then Sum l  = o by A0,TSum; then
   o in the set of all Sum(l) where l is Linear_Combination of Base(BE,BK);
   hence o in the carrier of Lin Base(BE,BK) by VECTSP_7:def 2;
   end;
now let o be object;
   assume A: o in the carrier of Lin Base(BE,BK);
   the carrier of Lin Base(BE,BK) = the set of all Sum(l) where
           l is Linear_Combination of Base(BE,BK) by VECTSP_7:def 2;
   then consider l1 being Linear_Combination of Base(BE,BK) such that
   B: o = Sum l1 by A;
   thus o in the carrier of V by B;
   end;
hence thesis by A,TARSKI:2,VECTSP_4:31;
end;
