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 Th12:
  for L being Linear_Combination of V st Carrier(L) c= the carrier
of W holds ex K being Linear_Combination of W st Carrier(K) = Carrier(L) & Sum(
  K) = Sum (L)
proof
  let L be Linear_Combination of V;
  assume
A1: Carrier(L) c= the carrier of W;
  then reconsider C = Carrier(L) as finite Subset of W;
  the carrier of W c= the carrier of V by RLSUB_1:def 2;
  then reconsider
  K = L|the carrier of W as Function of the carrier of W, REAL by FUNCT_2:32;
A2: K is Element of Funcs(the carrier of W, REAL) by FUNCT_2:8;
A3: dom K = the carrier of W by FUNCT_2:def 1;
  now
    let w be VECTOR of W;
A4: w is VECTOR of V by RLSUB_1:10;
    assume not w in C;
    then L.w = 0 by A4,RLVECT_2:19;
    hence K.w = 0 by A3,FUNCT_1:47;
  end;
  then reconsider K as Linear_Combination of W by A2,RLVECT_2:def 3;
  take K;
  thus thesis by A1,Th10;
end;
