theorem
  for W1, W2 being Submodule of V holds V
  is_the_direct_sum_of W1,W2 implies W2 is Linear_Compl of W1
  proof
    let W1, W2 be Submodule of V;
    assume V is_the_direct_sum_of W1,W2;
    then A1:V is_the_direct_sum_of W2,W1 by VECTSP_5:41;
    then W1 is with_Linear_Compl;
    hence thesis by Def19,A1;
  end;
