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  "body": "\n## Proof\n\nWe use the sign of the derivative.\n\n1. By [[the derivative of exp|0EX9GFJ0B3]], the [[exponential function|0RMVS7RXZX]] is differentiable on $\\R$, and $\\exp'(x) = \\exp(x)$ for all real $x$.\n2. [[Differentiable functions are continuous|07ZMX4ANJS]], so $\\exp$ is continuous on $\\R$.\n3. By [[the positivity of exp|0P9BNBW2W3]] and step 1, $\\exp'(x) = \\exp(x) \u003e 0$ for all real $x$.\n4. The interval $I = \\R$ has every point as an interior point. By steps 1 to 3, the theorem that a [[function with positive derivative is strictly increasing|07J5X2T1ZX]] applies to $\\exp$ on $\\R$.\n5. So $x \u003c y$ implies $\\exp(x) \u003c \\exp(y)$.\n\n## Idea\n\nThe slope of the exponential function at each point equals its value there, and this value is positive. A function whose slope is positive everywhere must rise, by the mean value theorem.\n",
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