theorem Th14:
  dom O = X & rng O = X
proof
  thus dom O = X
  proof
    thus dom O c= X;
    let x be object;
    assume x in X;
    then [x,x] in O by Th3;
    hence thesis by XTUPLE_0:def 12;
  end;
  thus rng O c= X;
  let x be object;
  assume x in X;
  then [x,x] in O by Th3;
  hence thesis by XTUPLE_0:def 13;
end;
