The proof is the "reason and logic" behind it. I understand what you're asking, but that's just not how math works. You want some pretty, intuitive reason for why it is true, but not every proof in math is elegant like that. Plenty of things just work becuase you show they do, and it's not always pretty, clear, or obvious. Sometimes you make a bunch of reductions and inferences, most of them seemingly completely unrelated, until you go from what you know to be true, to what you want to prove to be true. And it's not pretty, and it's hard to follow, and it challenges some people's notion of math being this elygant, aesthetically pleasing set of rules that are somehow hidden in the universe, and we have to discover them.

That's not how math works. It wasn't hidden in the world by God. Humans made it up, and sometimes, even if we start from something simple and pretty, and we end at something simple and pretty, the journey may be complicated and ugly. And we just have to deal with it.

So yeah, you can look up the proofs online, no need for someone to transcribe them here, but there's no guarantee you will like them. They are not meant to be elegant, they are just meant to be correct.