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  "body": "\n## Statement\n\nFor all real numbers $x$ and $y$,\n$$\\exp(x + y) = \\exp(x) \\exp(y).$$\n\n## Remarks\n\nMultiply the two series by the [[Cauchy product|0AN28XQ45Z]]. The binomial theorem gives $\\sum_{k=0}^{n} \\frac{x^k}{k!} \\frac{y^{n-k}}{(n-k)!} = \\frac{(x + y)^n}{n!}$.\nIn particular, $\\exp(x) \\exp(-x) = \\exp(0) = 1$.\nSo $\\exp$ is a group homomorphism from $(\\mathbb{R}, +)$ to the multiplicative group $(0, \\infty)$. Mathlib states the formula as `Real.exp_add`.\n\n## Explanations\n\n### L1 human\n\nThe exponential function turns addition into multiplication.\nThis is the rule $e^{x + y} = e^x e^y$ for powers.\n",
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