reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th8:
  for V be VectSp of K, W be Subspace of V, L be Linear_Combination of V
  holds L|the carrier of W is Linear_Combination of W
proof
  let V be VectSp of K;
  let W be Subspace of V;
  let L be Linear_Combination of V;
  set cW=the carrier of W;
  cW c=[#]V by VECTSP_4:def 2;
  then L|cW is Function of cW,the carrier of K by FUNCT_2:32;
  then reconsider L1=L|cW as Element of Funcs(cW,the carrier of K)
    by FUNCT_2:8;
  A1: for v be Element of W st not v in Carrier L/\cW holds L1.v=0.K
  proof
   let v be Element of W;
   reconsider w=v as Element of V by VECTSP_4:10;
   assume not v in Carrier L/\cW;
   then A2: not v in Carrier L by XBOOLE_0:def 4;
   L.w=L1.v by FUNCT_1:49;
   hence thesis by A2,VECTSP_6:2;
  end;
  Carrier L/\cW c=cW by XBOOLE_1:17;
  hence thesis by A1,VECTSP_6:def 1;
end;
