theorem
  for V being Z_Module, W being with_Linear_Compl Submodule of V,
  L being Linear_Compl of W, v being Vector of V,
  t being Element of [:the carrier of V, the carrier of V:]
  holds t`1 + t`2 = v & t`1 in W & t`2 in L implies t = v |-- (W,L)
  proof
    let V be Z_Module, W be with_Linear_Compl Submodule of V,
    L be Linear_Compl of W;
    V is_the_direct_sum_of W,L by Th124;
    hence thesis by VECTSP_5:def 6;
  end;
