reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;

theorem Th3:
  for F being FinSequence of the carrier of V st F is one-to-one
for L being Linear_Combination of V st Carrier(L) c= rng F holds Sum(L (#) F) =
  Sum(L)
proof
  let F be FinSequence of the carrier of V such that
A1: F is one-to-one;
  rng F c= rng F;
  then reconsider X = rng F as Subset of rng F;
  let L be Linear_Combination of V such that
A2: Carrier(L) c= rng F;
  consider G being FinSequence of the carrier of V such that
A3: G is one-to-one and
A4: rng G = Carrier(L) and
A5: Sum(L) = Sum(L (#) G) by VECTSP_6:def 6;
  reconsider A = rng G as Subset of rng F by A2,A4;
  set F1 = F - A`;
  X \ A` = X /\ A`` by SUBSET_1:13
    .= A by XBOOLE_1:28;
  then
A6: rng F1 = rng G by FINSEQ_3:65;
  F1 is one-to-one by A1,FINSEQ_3:87;
  then F1, G are_fiberwise_equipotent by A3,A6,RFINSEQ:26;
  then
A7: ex Q being Permutation of dom G st F1 = G*Q by RFINSEQ:4;
  reconsider F1, F2 = F - A as FinSequence of the carrier of V by FINSEQ_3:86;
  rng F2 = rng F \ rng G by FINSEQ_3:65;
  then
A8: rng F2 misses rng G by XBOOLE_1:79;
  ex P being Permutation of dom F st (F - A`) ^ (F - A) = F* P by FINSEQ_3:115;
  then Sum(L (#) F) = Sum(L (#) (F1^F2)) by Th1
    .= Sum((L (#) F1) ^ (L (#) F2)) by VECTSP_6:13
    .= Sum(L (#) F1) + Sum(L (#) F2) by RLVECT_1:41
    .= Sum(L (#) F1) + 0.V by A4,A8,Th2
    .= Sum(L (#) G) + 0.V by A7,Th1
    .= Sum(L) by A5,RLVECT_1:4;
  hence thesis;
end;
