
theorem Th19:
  for V being RealUnitarySpace, W being Subspace of V, K being
Linear_Combination of W holds ex L being Linear_Combination of V st Carrier(K)
  = Carrier(L) & Sum(K) = Sum(L)
proof
  let V be RealUnitarySpace;
  let W be Subspace of V;
  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);
  reconsider K9= K as Function of the carrier of W, REAL;
A1: the carrier of W c= the carrier of V by RUSUB_1:def 1;
  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 REAL & 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
A4:   not x in W;
     0 in REAL by XREAL_0:def 1;
      hence thesis by A4;
    end;
  end;
  ex L being Function of the carrier of V, REAL 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, REAL such that
A5: for x being object st x in the carrier of V holds P[x, L.x];
A6: 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] by
STRUCT_0:def 5;
    then P[v, K.v] & K.v = 0 or P[v, 0];
    hence L.v = 0 by A5;
  end;
  L is Element of Funcs(the carrier of V, REAL) by FUNCT_2:8;
  then reconsider L as Linear_Combination of V by A6,RLVECT_2:def 3;
  reconsider L9= L|the carrier of W as Function of the carrier of W, REAL by A1
,FUNCT_2:32;
  take L;
  now
    let x be object;
    assume x in Carrier(L) & not x in the carrier of W;
    then (ex v being VECTOR of V st x = v & L.v <> 0 )& not x in W by
STRUCT_0:def 5;
    hence contradiction by A5;
  end;
  then
A7: Carrier(L) c= the carrier of W;
  now
    let x be object;
    assume
A8: x in the carrier of W;
    then P[x, L.x] by A5,A1;
    hence K9.x = L9.x by A8,FUNCT_1:49,STRUCT_0:def 5;
  end;
  then K9 = L9 by FUNCT_2:12;
  hence thesis by A7,Th18;
end;
