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Formal Proof Elements

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Lemma

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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.

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Axiom

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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.

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Corollary

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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.

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Theorem

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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.

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Proof

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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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