
theorem
  for V being RealUnitarySpace, L being Linear_Combination of V, A being
  Subset of V st Carrier(L) c= the carrier of Lin(A) holds ex K being
  Linear_Combination of A st Sum(L) = Sum(K)
proof
  let V be RealUnitarySpace;
  let L be Linear_Combination of V, A be Subset of V;
  consider F being FinSequence of V such that
  F is one-to-one and
A1: rng F = Carrier(L) and
A2: Sum(L) = Sum(L (#) F) by RLVECT_2:def 8;
  assume Carrier(L) c= the carrier of Lin(A);
  then consider K being Linear_Combination of A such that
A3: Sum(L (#) F) = Sum(K) by A1,Th16;
  take K;
  thus thesis by A2,A3;
end;
