 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 Th7:
  X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X
  & a <> 0.INT.Ring & Sum KL1 = a * Sum KL2 implies KL1 = a * KL2
  proof
    assume that
    A1: X is linearly-independent & Carrier KL1 c= X and
    A2: Carrier KL2 c= X & a <> 0.INT.Ring & Sum(KL1) = a * Sum(KL2);
    Carrier(a * KL2) c= X & Sum(KL1) = Sum(a * KL2)
    by A2,ZMODUL02:29, ZMODUL02:53;
    hence thesis by A1,Th5;
  end;
