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 Th7:
  X is linearly-independent & Carrier KL1 c= X & Carrier KL2 c= X
  & a <> 0.K & 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.K & Sum(KL1) = a * Sum(KL2);
  Carrier(a * KL2) c= X & Sum(KL1) = Sum(a * KL2) by A2,VECTSP_6:29,45;
  hence thesis by A1,Th5;
end;
