
theorem
  for f be Relation holds dom f is complex-membered iff f is COMPLEX-defined
  proof
    let f be Relation;
    thus dom f is complex-membered implies f is COMPLEX-defined
    proof
      set E = (dom f)\/COMPLEX;
      reconsider X = dom f as Subset of E by XBOOLE_1:7;
      reconsider Y = COMPLEX as Subset of E by XBOOLE_1:7;
      assume dom f is complex-membered; then
      for x be Element of E st x in dom f holds x in COMPLEX
        by XCMPLX_0:def 2; then
      X c= Y by SUBSET_1:2;
      hence thesis by RELAT_1:def 18;
    end;
    thus thesis;
  end;
