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  "body": "\n## Proof\n\nWe show that $\\exp(x)$ is a square and is not zero.\n\n1. Let $\\exp$ be the [[exponential function|0RMVS7RXZX]]. At $0$, every term $\\frac{0^n}{n!}$ with $n \\ge 1$ is $0$, and the term with $n = 0$ is $1$. So $\\exp(0) = 1$.\n2. Let $x$ be real. By the [[exponential of a sum|0E8HNVBCSP]], $\\exp(x) = \\exp\\left(\\frac{x}{2} + \\frac{x}{2}\\right) = \\exp\\left(\\frac{x}{2}\\right)^2$. A square of a real number is nonnegative, so $\\exp(x) \\ge 0$.\n3. By the exponential of a sum and step 1, $\\exp(x) \\exp(-x) = \\exp(x + (-x)) = \\exp(0) = 1$.\n4. If $\\exp(x) = 0$, the left side of step 3 would be $0$, not $1$. So $\\exp(x) \\ne 0$.\n5. By steps 2 and 4, $\\exp(x) \u003e 0$ for every real $x$.\n\n## Idea\n\nThe functional equation writes $\\exp(x)$ as the square of $\\exp(x/2)$, so it is never negative. It also gives $\\exp(x) \\exp(-x) = 1$, so it is never zero.\n",
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