As the above poster says, the "true" bottom lines are sets of axioms ("non-logical" axioms, I believe). In mathematics, we "assume" these things to be true, and then derive the rest of mathematics from there... so mathematical statements can only be said to be "true" in the context of said axioms. We prove all our statements, but we have begun from the assumption of said axioms and hence the truth value of the statements is dependent on them. Some axioms may be "self-evident" truths (if there can be such a thing), but some are simply defining properties e.g. the existence of a null vector in the definition of vector spaces.
If we started from a completely different set of axioms (assuming there exists another such set from which it is actually possible to derive anything), I see no reason in principle that we couldn't derive completely new variations of mathematics (whether or not these variations would reflect current fields is something I wouldn't want to comment on). Some clever person somewhere may have somehow proven that this isn't true, but if so I don't know about it (though that isn't saying much).
But generally speaking, the level of proof needed is context dependent. If the audience for whom you are writing accepts the validity of a statement, there is typically no need to prove it. For example, in an exam your lecturer will typically have given you a bunch of theorems (whether explicitly or implicitly) which you are free to use without proof and so this is "where the buck stops", as it were, in this instance.
I think a schooling system in which every statement was proven from very first principles would be essentially impossible to implement... there is a reason pure mathematics is not usually introduced until university.