
theorem
  for f be Relation holds dom f is natural-membered iff f is NAT-defined
  proof
    let f be Relation;
    thus dom f is natural-membered implies f is NAT-defined
    proof
      set E = (dom f)\/NAT;
      reconsider X = dom f as Subset of E by XBOOLE_1:7;
      reconsider Y = NAT as Subset of E by XBOOLE_1:7;
      assume dom f is natural-membered; then
      for x be Element of E st x in dom f holds x in NAT
        by ORDINAL1:def 12; then
      X c= Y by SUBSET_1:2;
      hence thesis by RELAT_1:def 18;
    end;
    thus thesis;
  end;
