 reserve x, y, y1, y2 for set;
 reserve V for Z_Module;
 reserve u, v, w for Vector of V;
 reserve F, G, H, I for FinSequence of V;
 reserve W, W1, W2, W3 for Submodule of V;
 reserve KL1, KL2 for Linear_Combination of V;
 reserve X for Subset of V;

theorem Th3:
  X is linearly-independent & Carrier(KL1) c= X & Carrier(KL2) c= X
  & Sum(KL1) = Sum(KL2) implies KL1 = KL2
  proof
    assume A1: X is linearly-independent;
    assume A2: Carrier(KL1) c= X;
    assume Carrier(KL2) c= X; then
    A3: Carrier(KL1) \/ Carrier(KL2) c= X by A2,XBOOLE_1:8;
    assume Sum(KL1) = Sum(KL2);
    then Sum(KL1) - Sum(KL2) = 0.V by RLVECT_1:5; then
    A4: KL1 - KL2 is Linear_Combination of Carrier(KL1 - KL2) &
    Sum(KL1 - KL2) = 0.V by ZMODUL02:55,VECTSP_6:def 4;
    Carrier(KL1 - KL2) c= Carrier(KL1) \/ Carrier(KL2) by ZMODUL02:40; then
    A5: Carrier(KL1 - KL2) is linearly-independent
    by A1,A3,XBOOLE_1:1,ZMODUL02:56;
    now
      let v be Vector of V;
      not v in Carrier(KL1 - KL2) by A5,A4;
      then (KL1 - KL2).v = 0;
      then KL1.v - KL2.v = 0 by ZMODUL02:39;
      hence KL1.v = KL2.v;
    end;
    hence thesis;
  end;
