
theorem
  for f be Relation holds rng f is rational-membered iff f is RAT-valued
  proof
    let f be Relation;
    thus rng f is rational-membered implies f is RAT-valued
    proof
      set E = (rng f)\/RAT;
      reconsider X = rng f as Subset of E by XBOOLE_1:7;
      reconsider Y = RAT as Subset of E by XBOOLE_1:7;
      assume rng f is rational-membered; then
      for x be Element of E st x in rng f holds x in RAT by RAT_1:def 2; then
      X c= Y by SUBSET_1:2;
      hence thesis by RELAT_1:def 19;
    end;
    thus thesis;
  end;
