
theorem Th20:
  for V being RealUnitarySpace, W being Subspace of V, 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 V being RealUnitarySpace;
  let W being Subspace of V;
  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 RUSUB_1:def 1;
  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 RUSUB_1:3;
    assume not w in C;
    then L.w = 0 by A4;
    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,Th18;
end;
