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  "body": "\n## Statement\n\nLet $F$ and $K$ be [[complete ordered fields|0A9E6X0S3E]].\nThen there is exactly one field isomorphism $\\varphi \\colon F \\to K$ that preserves the order, that is, with $\\varphi(a) \\le \\varphi(b)$ whenever $a \\le b$.\n\n## Remarks\n\nThe proof first maps the rational elements of $F$ to those of $K$. Each other element is the supremum of the rational elements below it, and $\\varphi$ maps it to the supremum of their images.\nIn fact each field isomorphism between two complete ordered fields preserves the order, because the nonnegative elements are exactly the squares.\n\n## Explanations\n\n### L1 human\n\nTwo people may build the real numbers in different ways, for example with Dedekind cuts and with Cauchy sequences.\nThis theorem says that they get the same number system, with only different names for the numbers.\n",
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