theorem
  for W being with_Linear_Compl Submodule of V, L being
  Linear_Compl of W holds W + L = the ModuleStr of 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 = the ModuleStr of V;
  end;
