
theorem Th16:
  for V being RealUnitarySpace, L being Linear_Combination of V, A
  being Subset of V, F being FinSequence 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 V be RealUnitarySpace;
  let L be Linear_Combination of V;
  let A be Subset of V;
  defpred P[Nat] means
for F being FinSequence 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);
  let F be FinSequence of V;
  assume
A1: rng F c= the carrier of Lin(A);
A2: len F is Nat;
A3: for n being Nat st P[n] holds P[n + 1]
  proof
    let n be Nat;
    assume
A4: P[n];
    let F be FinSequence of V such that
A5: rng F c= the carrier of Lin(A) and
A6: len F = n + 1;
    consider G being FinSequence, x being object such that
A7: F = G^<*x*> by A6,FINSEQ_2:18;
    reconsider G, x9= <*x*> as FinSequence of V by A7,FINSEQ_1:36;
A8: rng(G^<*x*>) = rng G \/ rng <*x*> by FINSEQ_1:31
      .= rng G \/ {x} by FINSEQ_1:38;
    then
A9: rng G c= rng F by A7,XBOOLE_1:7;
    {x} c= rng F by A7,A8,XBOOLE_1:7;
    then {x} c= the carrier of Lin(A) by A5;
    then x in {x} implies x in the carrier of Lin(A);
    then
A10: x in Lin(A) by STRUCT_0:def 5,TARSKI:def 1;
    then
A11: x in V by RUSUB_1:2;
    consider LA1 being Linear_Combination of A such that
A12: x = Sum(LA1) by A10,Th1;
    reconsider x as VECTOR of V by A11,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
A13: Sum(L (#) G) = Sum(LA2) by A4,A5,A6,A7,A9,XBOOLE_1:1;
    L.x * LA1 is Linear_Combination of A by RLVECT_2:44;
    then
A14: LA2 + L.x * LA1 is Linear_Combination of A by RLVECT_2:38;
    Sum(L (#) F) = Sum((L (#) G) ^ (L (#) x9)) by A7,RLVECT_3:34
      .= Sum(LA2) + Sum(L (#) x9) by A13,RLVECT_1:41
      .= Sum(LA2) + Sum(<*L.x * x*>) by RLVECT_2:26
      .= Sum(LA2) + L.x * Sum(LA1) by A12,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 A14;
  end;
A15: P[0]
  proof
    let F be FinSequence of V;
    assume that
    rng F c= the carrier of Lin(A) and
A16: len F = 0;
    F = <*>(the carrier of V) by A16;
    then L (#) F = <*>(the carrier of V) by RLVECT_2:25;
    then
A17: 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 A17;
  end;
  for n being Nat holds P[n] from NAT_1:sch 2(A15, A3);
  hence thesis by A1,A2;
end;
