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 Th5:
  for L being Linear_Combination of V for F being FinSequence of
  the carrier of V st Carrier(L) misses rng F holds Sum(L (#) F) = 0.V
proof
  let L be Linear_Combination of V;
  defpred P[FinSequence] means for G being FinSequence of the carrier of V st
  G = $1 holds Carrier(L) misses rng G implies Sum(L (#) G) = 0.V;
A1: for p being FinSequence, x being object st P[p] holds P[p^<*x*>]
  proof
    let p be FinSequence, x be object such that
A2: P[p];
    let G be FinSequence of the carrier of V;
    assume
A3: G = p^<*x*>;
    then reconsider p, x9= <*x*> as FinSequence of the carrier of V by
FINSEQ_1:36;
    x in {x} by TARSKI:def 1;
    then
A4: x in rng x9 by FINSEQ_1:38;
    rng x9 c= the carrier of V by FINSEQ_1:def 4;
    then reconsider x as VECTOR of V by A4;
    assume Carrier(L) misses rng G;
    then
A5: {} = Carrier(L) /\ rng G by XBOOLE_0:def 7
      .= Carrier(L) /\ (rng p \/ rng<*x*>) by A3,FINSEQ_1:31
      .= Carrier(L) /\ (rng p \/ {x}) by FINSEQ_1:38
      .= Carrier(L) /\ rng p \/ Carrier(L) /\ {x} by XBOOLE_1:23;
    then Carrier(L) /\ rng p = {};
    then Carrier(L) misses rng p by XBOOLE_0:def 7;
    then
A6: Sum(L (#) p) = 0.V by A2;
A7: Carrier(L) /\ {x} = {} by A5;
    now
A8:   x in {x} by TARSKI:def 1;
      assume x in Carrier(L);
      hence contradiction by A7,A8,XBOOLE_0:def 4;
    end;
    then
A9: L.x = 0 by RLVECT_2:19;
    Sum(L (#) G) = Sum((L (#) p) ^ (L (#) x9)) by A3,RLVECT_3:34
      .= Sum(L (#) p) + Sum(L (#) x9) by RLVECT_1:41
      .= 0.V + Sum(<* L.x * x *>) by A6,RLVECT_2:26
      .= Sum(<* L.x * x *>) by RLVECT_1:4
      .= 0 * x by A9,RLVECT_1:44
      .= 0.V by RLVECT_1:10;
    hence thesis;
  end;
A10: P[{}]
  proof
    let G be FinSequence of the carrier of V;
    assume G = {};
    then
A11: L (#) G = <*>(the carrier of V) by RLVECT_2:25;
    assume Carrier(L) misses rng G;
    thus thesis by A11,RLVECT_1:43;
  end;
A12: for p being FinSequence holds P[p] from FINSEQ_1:sch 3(A10, A1);
  let F be FinSequence of the carrier of V;
  assume Carrier(L) misses rng F;
  hence thesis by A12;
end;
