 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;

theorem Th6:
  X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X
  & Carrier KL3 c= X & Sum KL1 = Sum KL2 + Sum KL3 implies KL1 = KL2 + KL3
  proof
    assume that
    A1: X is linearly-independent & Carrier KL1 c= X and
    A2: Carrier KL2 c= X & Carrier KL3 c= X and
    A3: Sum KL1 = Sum(KL2) + Sum(KL3);
    Carrier(KL2 + KL3) c= Carrier(KL2) \/ Carrier(KL3) &
    Carrier(KL2) \/ Carrier (KL3) c= X by A2,ZMODUL02:26,XBOOLE_1:8;
    then
    A4: Carrier(KL2 + KL3) c= X;
    Sum(KL1) = Sum(KL2 + KL3) by A3,ZMODUL02:52;
    hence thesis by A1,A4,Th5;
  end;
