
theorem
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) holds lift(down l,BE) = 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 l be Linear_Combination of Base(BE,BK);
H4: Carrier l c= Base(BE,BK) by VECTSP_6:def 4;
now let o be object;
  assume AS: o in the carrier of VecSp(K,F); then
  reconsider c = o as Element of K by FIELD_4:def 6;
  reconsider cV = o as Element of VecSp(K,F) by AS;
  per cases;
  suppose AS: c in Base(BE,BK); then
    consider a,b being Element of K such that
    A: c = a * b & a in BE & b in BK;
    B: (down l).b = Sum down(l,b) by A,down2;
    consider l2 being Linear_Combination of BE such that
    D: Sum l2 = (down l).b &
       for a being Element of K st a in BE & a * b in Base(BE,BK)
       holds lift(down l,BE).(a*b) = l2.a by A,lif;
    l2 = down(l,b) by D,B,ZMODUL033;
    then lift(down l,BE).(a*b)
       = down(l,b).a by A,D,AS
      .= l.(a*b) by A,down1;
    hence l.o = lift(down l,BE).o by A;
    end;
  suppose X: not c in Base(BE,BK); then
    A: l.cV = 0.F by H4,VECTSP_6:2;
    Carrier lift(down l,BE) c=  Base(BE,BK) by VECTSP_6:def 4;
    hence l.o = lift(down l,BE).o by A,X,VECTSP_6:2;
    end;
  end;
hence thesis;
end;
