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
  for U be Subspace of n-VectSp_over F_Real, W be Subspace of TOP-REAL n
  for LU be Linear_Combination of U, LW be Linear_Combination of W st LU = LW
  holds Carrier LU = Carrier LW & Sum LU = Sum LW
proof
  set V=n-VectSp_over F_Real;
  set T=TOP-REAL n;
  let U be Subspace of V,W be Subspace of TOP-REAL n;
  let LU be Linear_Combination of U,LW be Linear_Combination of W such that
   A1: LU=LW;
  reconsider LW9=LW as Function of the carrier of W,REAL;
  defpred P[object,object] means
   ($1 in W & $2=LW.$1) or(not$1 in W & $2=In(0,REAL));
  A2: dom LU=[#]U & dom LW=[#]W by FUNCT_2:def 1;
  A3: for x be object st x in the carrier of T
ex y be object st y in REAL & P[x,y]
  proof
   let x be object;
   assume x in the carrier of T;
   then reconsider x as VECTOR of T;
   per cases;
   suppose A4: x in W;
    then reconsider x as VECTOR of W;
    P[x,LW.x] by A4;
    hence thesis;
   end;
   suppose not x in W;
    hence thesis;
   end;
  end;
  consider L be Function of the carrier of T,REAL such that
   A5: for x be object st x in the carrier of T holds P[x,L.x]
from FUNCT_2:sch 1
(A3);
  A6: the carrier of W c=the carrier of T by RLSUB_1:def 2;
  then reconsider C=Carrier(LW) as finite Subset of T by XBOOLE_1:1;
  A7: L is Element of Funcs(the carrier of T,REAL) by FUNCT_2:8;
  now let v be VECTOR of T;
   assume not v in C;
   then P[v,LW.v] & not v in C & v in the carrier of W or P[v,0]
     by STRUCT_0:def 5;
   then P[v,LW.v] & LW.v=0 or P[v,0] by RLVECT_2:19;
   hence L.v=0 by A5;
  end;
  then reconsider L as Linear_Combination of T by A7,RLVECT_2:def 3;
  reconsider L9=L|the carrier of W as Function of the carrier of W,REAL
    by A6,FUNCT_2:32;
  now let x be object;
   assume A8: x in the carrier of W;
   then P[x,L.x] by A6,A5;
   hence LW9.x=L9.x by A8,FUNCT_1:49,STRUCT_0:def 5;
  end;
  then A9: LW=L9 by FUNCT_2:12;
  reconsider K=L as Linear_Combination of V by Th1;
  now let x be object;
   assume that
    A10: x in Carrier(L) and
    A11: not x in the carrier of W;
   consider v being VECTOR of T such that
    A12: x=v and
    A13: L.v<>0 by A10,RLVECT_5:3;
   P[v,0] by A11,A12,STRUCT_0:def 5;
   hence contradiction by A5,A13;
  end;
  then A14: Carrier(L)c=the carrier of W;
  then A15: Carrier(L)=Carrier(LW) & Sum(L)=Sum(LW) by A9,RLVECT_5:10;
  A16: Carrier(L)=Carrier(K) by Th2;
  then Sum K=Sum LU by A1,A2,A14,A9,VECTSP_9:7;
  hence thesis by A1,A2,A9,A15,A16,Th5,VECTSP_9:7;
end;
