
theorem DNP:
  for f be complex-valued Function holds f = delneg f - delpos f
  proof
    let f be complex-valued Function;
    dom (delneg f) = dom f & dom (delpos f) = dom f by DMN; then
    A1: dom f =  dom (delneg f) /\ dom (delpos f)
    .= dom ((delneg f) - (delpos f)) by VALUED_1:12;
    for x be object st x in dom f holds
    (delneg f).x - (delpos f).x = f.x by VAL;
    hence thesis by A1,VALUED_1:14;
  end;
