theorem
  R is empty-yielding iff for X st X in rng R holds X = {}
proof
  thus R is empty-yielding implies
  for X st X in rng R holds X = {} by TARSKI:def 1;
  assume
A1: for X st X in rng R holds X = {};
  let Y be object;
  assume Y in rng R;
  then Y = {} by A1;
  hence thesis by TARSKI:def 1;
end;
