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  "body": "\n## Definition\n\nFor a real number $x \\ge 0$, the square root $\\sqrt{x}$ is the unique real number $y \\ge 0$ with $y^2 = x$.\nThe square root function is the function $x \\mapsto \\sqrt{x}$ on $[0, \\infty)$.\n\n## Remarks\n\nThe number $y$ exists by the [[intermediate value theorem|00KAFBR32E]] for $t \\mapsto t^2$. It is unique because $t^2$ is [[strictly increasing|0TR9SR9FTB]] on $[0, \\infty)$.\nThe function is continuous and strictly increasing. It is differentiable on $(0, \\infty)$, but not at $0$.\nFor $x \u003e 0$, $\\sqrt{x} = x^{1/2}$. Also $\\sqrt{x^2} = |x|$ for every real $x$. Mathlib defines `Real.sqrt`, with $\\sqrt{x} = 0$ for $x \u003c 0$.\n\n## Explanations\n\n### L1 human\n\nThe square root of $x$ is the nonnegative number whose square is $x$.\nFor example, $\\sqrt{9} = 3$, not $-3$.\n",
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