
theorem Tlift2:
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 BK holds down lift(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 being Basis of VecSp(E,F), BK being Basis of VecSp(K,E);
let l1 being Linear_Combination of BK;
H1: the carrier of VecSp(K,E) = the carrier of K &
    the carrier of VecSp(E,F) = the carrier of E by FIELD_4:def 6;
E is Subring of K by FIELD_4:def 1; then
H3: the carrier of E c= the carrier of K by C0SP1:def 3;
H4: Carrier l1 c= BK by VECTSP_6:def 4;
now let o be object;
  assume AS: o in the carrier of VecSp(K,E); then
  reconsider b = o as Element of K by FIELD_4:def 6;
  reconsider bV = o as Element of VecSp(K,E) by AS;
  per cases;
  suppose X: b in BK; then
    consider l2 being Linear_Combination of BE such that
    A: Sum l2 = l1.b &
       for a being Element of K st a in BE & a * b in Base(BE,BK)
       holds lift(l1,BE).(a*b) = l2.a by lif;
    C: Carrier l2 c= BE by VECTSP_6:def 4;
    down(lift(l1,BE),b) = l2
      proof
      now let o be object;
        assume AS: o in the carrier of VecSp(E,F); then
        reconsider a = o as Element of K by H1,H3;
        reconsider aV = o as Element of VecSp(E,F) by AS;
        per cases;
        suppose Y: a in BE; then
          Z: a * b in Base(BE,BK) by X;
          down(lift(l1,BE),b).a = lift(l1,BE).(a*b) by X,Y,down1
                               .= l2.a by A,Y,Z;
          hence down(lift(l1,BE),b).o = l2.o;
          end;
        suppose AS1: not a in BE; then
          D: l2.aV = 0.F by C,VECTSP_6:2;
          Carrier down(lift(l1,BE),b) c= BE by VECTSP_6:def 4;
          hence down(lift(l1,BE),b).o = l2.o by D,AS1,VECTSP_6:2;
          end;
        end;
      hence thesis;
      end;
    hence (down lift(l1,BE)).o = l1.o by A,X,down2;
    end;
  suppose X: not b in BK; then
    A: l1.bV = 0.E by H4,VECTSP_6:2;
    Carrier down lift(l1,BE) c= BK by VECTSP_6:def 4;
    hence (down lift(l1,BE)).o = l1.o by A,X,VECTSP_6:2;
    end;
  end;
hence thesis;
end;
