theorem Th124:
  for W being with_Linear_Compl Submodule of V,
  L being Linear_Compl of W holds
  V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L
  proof
    let W be with_Linear_Compl Submodule of V,
    L be Linear_Compl of W;
    thus V is_the_direct_sum_of L,W by Def19;
    hence thesis by VECTSP_5:41;
  end;
