A Successor of the Apostles is a man appointed by the Roman Pontiff to govern a particular church (i.e. a diocese). He has the power to teach, ex officio, and the power to rule (i.e. jurisdiction).
Now you have resorted to just baldly stating what you can't even come close to proving. Weak, really weak.
Find me ANYONE, any theologian, any Pope, who defines a successor of the Apostles that way, besides your Quigley. Tell me where the Church has ever defined what a Successor of the Apostles exactly is.
Here, let me try your method: "A successor of the Apostles is someone with valid Holy Orders." There, I said it, it must be true.
What must be determined are the minimum qualifications for being a successor of the Apostles. Is it being able to trace your Holy Orders back to the Apostles, or is it being able to teach and to rule, having been sent by the Pope? Proof, SJB, proof.
Also, not everyone agrees with you about the Eastern schismatics not being successors of the Apostles, read the FishEaters thread about it. You are trying to make this sound so certain when almost no sede or SSPX priests ( who also lack ordinary jurisdiction ) I know of are as certain as you are, when almost no one on the Internet is as certain. Are you more educated?
I have already proved you used quotes that didn't pertain to the question. I don't care for how you're trying to bully this one through, the unwary could fall into an error like Gerry Matatics, thinking they can't go to any priest because they're all illicit.