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 Th9:
  for L being Linear_Combination of V for 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 L be Linear_Combination of V, A be Subset of V;
  consider F being FinSequence of the carrier 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,Th8;
  take K;
  thus thesis by A2,A3;
end;
