reserve k,t,i,j,m,n for Nat,
  x,y,y1,y2 for object,
  D for non empty set;
reserve K for Field,
  V for VectSp of K,
  a for Element of K,
  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,VECTSP_6:23,XBOOLE_1:8;
  then
A4: Carrier(KL2 + KL3) c= X;
  Sum(KL1) = Sum(KL2 + KL3) by A3,VECTSP_6:44;
  hence thesis by A1,A4,Th5;
end;
