
theorem for f be real-valued Function holds delall f = (delneg f)(#)(delpos f)
  proof
    let f be real-valued Function;
    dom (delneg f) = dom f & dom (delpos f) = dom f by DMN; then
    A1: dom ((delneg f)(#)(delpos f)) = (dom f) /\ (dom f) by VALUED_1:def 4
    .= dom (delall f) by DMN;
    for k be object st k in dom (delall f) holds
      ((delneg f)(#)(delpos f)).k = (delall f).k;
    hence thesis by A1,FUNCT_1:2;
  end;
