A theorem is a statement in mathematics that has been proven to be true within the framework of a given mathematical system. The proof of a theorem is a logical argument demonstrating that the theorem follows from the axioms of the system, as well as any previously established theorems.

The process of proving a theorem involves a rigorous application of deductive reasoning. This means that the truth of the theorem is conditional on the truth of the axioms (or postulates) and the correctness of the reasoning in the proof.

An axiom, in mathematics and logic, is a statement or proposition that is regarded as being self-evidently true, without the need for proof. Axioms serve as the starting points for developing a mathematical theory. Different sets of axioms can give rise to different, but consistent, mathematical systems.

A postulate, also known as an axiom, is a statement that is assumed to be true without proof. Postulates are the foundational assumptions upon which a mathematical or logical system is built. They serve as the starting points from which other truths can be logically deduced.