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  "body": "\n## Summary\n\nA mathematical proof is an argument that shows that a statement follows from axioms, definitions and statements that are already proved.\nEach step of a proof follows from earlier steps by a valid rule of inference, such as [[modus ponens|0Y0MQYST0M]].\nA proved statement is a theorem.\n\n## Remarks\n\nCommon methods are [[direct proof|0NTF6DMVAP]], [[proof by contrapositive|0BB5TPGQ1N]], [[proof by contradiction|0J0XBKG64S]] and [[proof by cases|0WCP0PVET6]].\nA formal proof is a sequence of formulas in a formal system. A computer can check it, for example in Lean.\nExamples, however many, do not prove a universal statement.\n\n## Explanations\n\n### L1 human\n\nA proof is a chain of small steps, and each step is certainly correct.\nIt starts from things that you accept or already know, and it ends at the statement.\nAfter a correct proof, the statement is true in every case, not only in the cases that you tested.\n",
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