 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem
  for F being FinSequence 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) :::by VECTSP_9:3;
  proof
    let F be FinSequence 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 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
    A7: ex Q being Permutation of dom G st F1 = G*Q
    by A3,A6,RFINSEQ:4,26;
    reconsider F1, F2 = F - A as FinSequence of V by FINSEQ_3:86;
    A8: rng F2 /\ rng G = (rng F \ rng G) /\ rng G by FINSEQ_3:65
    .= {} by XBOOLE_0:def 7,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 Th5
    .= Sum((L (#) F1) ^ (L (#) F2)) by ZMODUL02:51
    .= Sum(L (#) F1) + Sum(L (#) F2) by RLVECT_1:41
    .= Sum(L (#) F1) + 0.V by A4,A8,Th6,XBOOLE_0:def 7
    .= Sum(L (#) G) + 0.V by A7,Th5
    .= Sum(L) by A5,RLVECT_1:4;
    hence thesis;
  end;
