reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th1:
  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) = Sum(KL1) + -Sum(KL1) by RLVECT_1:def 11
    .= 0.V by RLVECT_1:5;
  then
A4: KL1 - KL2 is Linear_Combination of Carrier(KL1 - KL2) & Sum(KL1 - KL2)
  = 0.V by RLVECT_2:def 6,RLVECT_3:4;
  Carrier(KL1 - KL2) c= Carrier(KL1) \/ Carrier(KL2) by RLVECT_2:55;
  then
A5: Carrier(KL1 - KL2) is linearly-independent by A1,A3,RLVECT_3:5,XBOOLE_1:1;
  now
    let v be VECTOR of V;
    not v in Carrier(KL1 - KL2) by A5,A4,RLVECT_3:def 1;
    then (KL1 - KL2).v = 0 by RLVECT_2:19;
    then KL1.v - KL2.v = 0 by RLVECT_2:54;
    hence KL1.v = KL2.v;
  end;
  hence thesis by RLVECT_2:def 9;
end;
