Maybe even easier example is the commutativity of multiplication itself.

That's a good point! I avoided that example because there's a pretty easy and convincing "proof" of the commutativity of multiplication, namely that turning a rectangle on its side doesn't change how many things constitute it So, it doesn't matter whether you count how many are in each row and then count how many rows there are, or if you do that with columns instead.

I think it's terribly sad that they don't encourage children to notice that or something like it. But there are a lot of things about education I find terribly sad and that I'm doing my damnest to fix.

But that doesn't seem much relevant to the question of truth of mathematical theorems. Whatever intuitive thought had lead to its discovery, people will agree that it is valid iff there is a formal proof.

Agreed, though there's no objective definition of what constitutes a "formal proof". Despite what it might seem like from the outside, there's no one axiomatic system and deductive set of rules to which all subfields of mathematics pay homage.