That's the point of the proof, it's to show you how things you already accept are true entail something you're less certain about. But perhaps the deeper point

That's the point of the proof, it's to show you how things you already accept are true entail something you're less certain about. But perhaps the deeper point is this is a lesson for children or students and has a level of rigor appropriate for, and not unusual in, pedagogy, not a ironclad argument for "serious" debates with internet contrarians.