theorem
  for V being Z_Module, W being with_Linear_Compl Submodule of V,
  L being Linear_Compl of W holds W is Linear_Compl of L
  proof
    let V be Z_Module, W be with_Linear_Compl Submodule of V,
    L be Linear_Compl of W;
    V is_the_direct_sum_of L,W by Def19; then
A1: V is_the_direct_sum_of W,L by VECTSP_5:41;
    then L is with_Linear_Compl;
    hence thesis by Def19,A1;
  end;
