reserve GF for Field,
  V for VectSp of GF,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n, m for Nat;

theorem Th8:
  for K being Linear_Combination of W ex L being
  Linear_Combination of V st Carrier(K) = Carrier(L) & Sum(K) = Sum (L)
proof
  let K be Linear_Combination of W;
  defpred P[object,object] means
   ($1 in W & $2 = K.$1) or (not $1 in W & $2 = 0.GF);
  reconsider K9= K as Function of the carrier of W, the carrier of GF;
A1: the carrier of W c= the carrier of V by VECTSP_4:def 2;
  then reconsider C = Carrier(K) as finite Subset of V by XBOOLE_1:1;
A2: for x being object st x in the carrier of V
   ex y being object st y in the carrier of GF & P[x, y]
  proof
    let x be object;
    assume x in the carrier of V;
    then reconsider x as Vector of V;
    per cases;
    suppose
A3:   x in W;
      then reconsider x as Vector of W by STRUCT_0:def 5;
      P[x, K.x] by A3;
      hence thesis;
    end;
    suppose
      not x in W;
      hence thesis;
    end;
  end;
  ex L being Function of the carrier of V, the carrier of GF st
   for x being object st x
  in the carrier of V holds P[x, L.x] from FUNCT_2:sch 1(A2);
  then consider
  L being Function of the carrier of V, the carrier of GF such that
A4: for x being object st x in the carrier of V holds P[x, L.x];
A5: now
    let v be Vector of V;
    assume not v in C;
    then P[v, K.v] & not v in C & v in the carrier of W or P[v, 0.GF] by
STRUCT_0:def 5;
    then P[v, K.v] & K.v = 0.GF or P[v, 0.GF] by VECTSP_6:2;
    hence L.v = 0.GF by A4;
  end;
  L is Element of Funcs(the carrier of V, the carrier of GF) by FUNCT_2:8;
  then reconsider L as Linear_Combination of V by A5,VECTSP_6:def 1;
  reconsider L9= L|the carrier of W as Function of the carrier of W, the
  carrier of GF by A1,FUNCT_2:32;
  take L;
  now
    let x be object;
    assume that
A6: x in Carrier(L) and
A7: not x in the carrier of W;
    consider v being Vector of V such that
A8: x = v and
A9: L.v <> 0.GF by A6,VECTSP_6:1;
    P[v, 0.GF] by A7,A8,STRUCT_0:def 5;
    hence contradiction by A4,A9;
  end;
  then
A10: Carrier(L) c= the carrier of W;
  now
    let x be object;
    assume
A11: x in the carrier of W;
    then P[x, L.x] by A4,A1;
    hence K9.x = L9.x by A11,FUNCT_1:49,STRUCT_0:def 5;
  end;
  then K9 = L9 by FUNCT_2:12;
  hence thesis by A10,Th7;
end;
