reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th8:
  for L being Linear_Combination of V for A being Subset of V for F
  being FinSequence of the carrier of V st rng F c= the carrier of Lin(A) holds
  ex K being Linear_Combination of A st Sum(L (#) F) = Sum(K)
proof
  let L be Linear_Combination of V;
  let A be Subset of V;
  defpred P[Nat] means
for F being FinSequence of the carrier of V
  st rng F c= the carrier of Lin(A) & len F = $1 holds ex K being
  Linear_Combination of A st Sum(L (#) F) = Sum(K);
A1: for n being Nat st P[n] holds P[n + 1]
  proof
    let n be Nat;
    assume
A2: P[n];
    let F be FinSequence of the carrier of V such that
A3: rng F c= the carrier of Lin(A) and
A4: len F = n + 1;
    consider G being FinSequence, x being object such that
A5: F = G^<*x*> by A4,FINSEQ_2:18;
    reconsider G, x9= <*x*> as FinSequence of the carrier of V by A5,
FINSEQ_1:36;
A6: rng(G^<*x*>) = rng G \/ rng <*x*> by FINSEQ_1:31
      .= rng G \/ {x} by FINSEQ_1:38;
    then
A7: rng G c= rng F by A5,XBOOLE_1:7;
    {x} c= rng F by A5,A6,XBOOLE_1:7;
    then {x} c= the carrier of Lin(A) by A3;
    then x in {x} implies x in the carrier of Lin(A);
    then
A8: x in Lin(A) by STRUCT_0:def 5,TARSKI:def 1;
    then consider LA1 being Linear_Combination of A such that
A9: x = Sum(LA1) by RLVECT_3:14;
    x in V by A8,RLSUB_1:9;
    then reconsider x as VECTOR of V by STRUCT_0:def 5;
    len(G^<*x*>) = len G + len <*x*> by FINSEQ_1:22
      .= len G + 1 by FINSEQ_1:39;
    then consider LA2 being Linear_Combination of A such that
A10: Sum(L (#) G) = Sum(LA2) by A2,A3,A4,A5,A7,XBOOLE_1:1;
    L.x * LA1 is Linear_Combination of A by RLVECT_2:44;
    then
A11: LA2 + L.x * LA1 is Linear_Combination of A by RLVECT_2:38;
    Sum(L (#) F) = Sum((L (#) G) ^ (L (#) x9)) by A5,RLVECT_3:34
      .= Sum(LA2) + Sum(L (#) x9) by A10,RLVECT_1:41
      .= Sum(LA2) + Sum(<*L.x * x*>) by RLVECT_2:26
      .= Sum(LA2) + L.x * Sum(LA1) by A9,RLVECT_1:44
      .= Sum(LA2) + Sum(L.x * LA1) by RLVECT_3:2
      .= Sum(LA2 + L.x * LA1) by RLVECT_3:1;
    hence thesis by A11;
  end;
  let F be FinSequence of the carrier of V;
  assume
A12: rng F c= the carrier of Lin(A);
A13: len F is Element of NAT;
A14: P[0]
  proof
    let F be FinSequence of the carrier of V;
    assume that
    rng F c= the carrier of Lin(A) and
A15: len F = 0;
    F = <*>(the carrier of V) by A15;
    then L (#) F = <*>(the carrier of V) by RLVECT_2:25;
    then
A16: Sum(L (#) F) = 0.V by RLVECT_1:43
      .= Sum(ZeroLC(V)) by RLVECT_2:30;
    ZeroLC(V) is Linear_Combination of A by RLVECT_2:22;
    hence thesis by A16;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A14, A1);
  hence thesis by A12,A13;
end;
