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Formal Proof Elements
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Lemma
A lemma is a proven statement used to help prove a theorem or solve a problem. Often considered as a 'helper theorem', it simplifies complex proofs by allowing the main proof to reference the lemma's conclusion.
Axiom
An axiom is a statement that is assumed to be true and serves as a starting point for further reasoning and arguments. Axioms form the basic building blocks from which theorems are derived.
Corollary
A corollary is a statement that follows readily from a previously proven statement or theorem. The role of a corollary in a proof is to present additional results that are less significant or are immediate consequences of a theorem without requiring a separate, in-depth proof.
Theorem
A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and axioms. The role of a theorem in a proof is to provide a conclusive assertion about a mathematical concept that has been rigorously established.
Proof
A proof is a logical argument that demonstrates the truth of a theorem. The role of a proof is to derive the theorem from axioms and previously established theorems using accepted logical steps, assuring the mathematical community of its validity.
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