theorem Th39:
  for L being Linear_Compl of W holds W + L = the ModuleStr of V &
  L + W = the ModuleStr of V
proof
  let L be Linear_Compl of W;
  V is_the_direct_sum_of W,L by Th38;
  hence W + L = the ModuleStr of V;
  hence thesis by Lm1;
end;
