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  "body": "\n## Definition\n\nLet $(P, \\le)$ be a [[partially ordered set|0QV2JS9RJ3]].\nThe set $P$ is *partially well-ordered* when for every infinite [[sequence|0G9EKM0E15]] $x_1, x_2, x_3, \\ldots$ in $P$ there are indices $i \u003c j$ with $x_i \\le x_j$.\nMore generally, let $S$ be a subset of a preordered set $(X, \\le)$.\nThe subset $S$ is partially well-ordered when every infinite sequence in $S$ has indices $i \u003c j$ with $x_i \\le x_j$.\n\n## Remarks\n\nSo a partially well-ordered set is a partial order that is a [[well-quasi-ordering|NH25G6GC07]].\n\nExample: $\\N^2$ with the product order.\nCheck: from any sequence $(a_n, b_n)$ extract a subsequence on which $a_n$ is non-decreasing, then a further subsequence on which $b_n$ is non-decreasing; two terms of it give $i \u003c j$ with $x_i \\le x_j$.\n\nNon-example: $\\N$ with the trivial order, where $x \\le y$ only when $x = y$, is a partial order but not partially well-ordered: the sequence $0, 1, 2, \\ldots$ has no $i \u003c j$ with $x_i \\le x_j$.\n\nEvery well-ordered set is partially well-ordered.\nIn Mathlib the subset form is `Set.IsPWO`.\n",
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