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 V be VectSp of K, A be linearly-independent Subset of V
  for L1,L2 be Linear_Combination of V st
     Carrier L1 c= A & Carrier L2 c= A & Sum L1 = Sum L2
  holds L1 = L2
proof
  let V be VectSp of K;
  let A be linearly-independent Subset of V;
  let L1,L2 be Linear_Combination of V such that
   A1: Carrier L1 c=A & Carrier L2 c=A and
   A2: Sum L1=Sum L2;
  (L1 is Linear_Combination of A) & L2 is Linear_Combination of A
    by A1,VECTSP_6:def 4;
  then A3: L1-L2 is Linear_Combination of A by VECTSP_6:42;
  Sum(L1-L2)=Sum L1-Sum L2 by VECTSP_6:47
   .=0.V by A2,RLVECT_1:15;
  then Carrier(L1-L2)={} by A3,VECTSP_7:def 1;
  then ZeroLC(V)=L1-L2 by VECTSP_6:def 3
   .=L1+-L2 by VECTSP_6:def 11
   .=-L2+L1 by VECTSP_6:25;
  then L1=--L2 by VECTSP_6:37;
  hence thesis;
end;
