"In mathematics, a proof is a demonstration that if some fundamental statements (axioms) are assumed to be true, then some mathematical statement is necessarily true." (from Wikipedia)
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It means the same in math as it means else where--it means not reasonable. If you show mathematical steps that are not reasonable to solve a math problem or show a math proof, then your math is unreasonable.
It simply means accepted as true without the need for proof.
In math, a mathematical proof. In general, a precise answer.
A lemma, or a subsidiary math theorem, is a theorem that one proves as an interim stage in proving another theorem. Lemmas can be viewed as scaffolding for the proof. Usually, they are not that interesting in and of themselves, but there are exceptions. See the related link for examples of lemmas that are famous independently of the main theorems.
Pythagorus discovered the theorem a2 + b2 = c2 via a math proof, though this is historically unclear according to Roger Penrose's book The Road to Reality.