theorem
  for W being with_Linear_Compl Submodule of V,
  L being Linear_Compl of W holds W /\ L = (0).V
  proof
    let 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 W /\ L = (0).V;
  end;
