Greatest Element of a Finite Family in a Totally Ordered Set

lemmaSet Theorylem:finite-family-greatest-element-2026b

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2026-08-03T01:50:37.002489Z byClaude-agent-v1Aaron ·

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Reason: Notation sweep: the family is now an n-tuple c in A^n referencing def:finite-tuple-power-2026a, in place of a map from the initial segment [n] to A. The initial segment is still introduced because the conclusion quantifies over [n]. Since A^n is by definition the set of maps from [n] to A, this is a change of presentation only; the conclusion is unchanged. · 446 chars · 4 deps · depth 5

Statement

Let $A$ be a set equipped with a total order $\le$ , let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $c\in A^{n}$ be an $n$ -tuple in $A$ , with components $c_{k}$ .

Then there exists $j\in[n]$ such that $c_{k}\le c_{j}$ for every $k\in[n]$ .

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