reserve N,M,K for ExtNat;
reserve X for ext-natural-membered set;

theorem Th2:
  for f being Function holds f is Ordinal-yielding iff
    for x being object st x in dom f holds f.x is Ordinal
proof
  let f be Function;
  hereby
    assume f is Ordinal-yielding;
    then consider A being Ordinal such that
      A1: rng f c= A by ORDINAL2:def 4;
    let x be object;
    assume x in dom f;
    then f.x in rng f by FUNCT_1:3;
    hence f.x is Ordinal by A1;
  end;
  assume A2: for x being object st x in dom f holds f.x is Ordinal;
  now
    reconsider A = sup rng f as Ordinal;
    take A;
    now
      let y be object;
      assume A3: y in rng f;
      then consider x being object such that
        A4: x in dom f & y = f.x by FUNCT_1:def 3;
      y is Ordinal by A2, A4;
      then A5: y in On rng f by A3, ORDINAL1:def 9;
      On rng f c= A by ORDINAL2:def 3;
      hence y in A by A5;
    end;
    hence rng f c= A by TARSKI:def 3;
  end;
  hence thesis by ORDINAL2:def 4;
end;
