reserve V,W for Z_Module;
reserve T for linear-transformation of V,W;
reserve T for linear-transformation of V,W;
reserve l for Linear_Combination of V;
reserve V,W for Z_Module;
reserve l for Linear_Combination of V;
reserve T for linear-transformation of V,W;

theorem
  for R being Ring
  for V being LeftMod of R, A being Subset of V,
  l1, l2 being Linear_Combination of A
  st Carrier(l1) misses Carrier(l2) holds
  Carrier(l1 - l2) = Carrier(l1) \/ Carrier(l2)
  proof
    let R be Ring;
    let V be LeftMod of R;
    let A be Subset of V,
    l1, l2 be Linear_Combination of A such that
    A1: Carrier(l1) misses Carrier(l2);
    A2: Carrier(l1) misses Carrier(-l2) by A1,VECTSP_6:38;
    thus Carrier(l1 - l2) = Carrier(l1 +- l2)
    .= Carrier(l1) \/ Carrier(-l2) by A2,Th31
    .= Carrier(l1) \/ Carrier(l2) by VECTSP_6:38;
  end;
