
theorem RV:
  for f be Relation holds rng f is real-membered iff f is real-valued
  proof
    let f be Relation;
    thus rng f is real-membered implies f is real-valued
    proof
      set E = (rng f)\/REAL;
      reconsider X = rng f as Subset of E by XBOOLE_1:7;
      reconsider Y = REAL as Subset of E by XBOOLE_1:7;
      assume rng f is real-membered; then
      for x be Element of E st x in rng f holds x in REAL
        by XREAL_0:def 1; then
      X c= Y by SUBSET_1:2;
      hence thesis by VALUED_0:def 3;
    end;
    thus thesis;
  end;
