theorem
  for V being Z_Module holds (0).V is Linear_Compl of (Omega).V &
  (Omega).V is Linear_Compl of (0).V
  proof
    let V be Z_Module;
    A1:  V is_the_direct_sum_of (0).V,(Omega).V &
    V is_the_direct_sum_of (Omega).V, (0).V by Th129;
    then
    A2: (Omega).V is with_Linear_Compl;
    (0).V is with_Linear_Compl by A1;
    hence thesis by Def19,A1,A2;
  end;
