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  "body": "\n## Definition\n\nLet $\\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]], let $S$ be the [[successor function|0832XVR512]] and let $+$ be the [[addition|0W5DSDFQ4G]] of $\\mathbb{N}$.\nMultiplication is the function $\\cdot \\colon \\mathbb{N} \\times \\mathbb{N} \\to \\mathbb{N}$, $(a, b) \\mapsto a \\cdot b$, with these two properties for all $a$ and $b$ in $\\mathbb{N}$:\n$$a \\cdot 0 = 0, \\qquad a \\cdot S(b) = a \\cdot b + a.$$\n\n## Remarks\n\nFor each $a$, these equations are a [[definition by recursion|0DSHAVH45F]] on $b$. Thus exactly one such function exists.\nFrom the definition, $a \\cdot 1 = a \\cdot 0 + a = a$.\nWe often write $ab$ for $a \\cdot b$.\n\n## Explanations\n\n### L1 human\n\nMultiplication is repeated addition: $3 \\cdot 4$ is $3 + 3 + 3 + 3 = 12$.\nThe definition says this in two rules: zero copies give $0$, and one more copy adds one more $a$.\n",
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