Smallest Element WRT Restricted Ordering
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Theorem
Let $\preceq$ be an ordering on $S$.
Let $T$ be a subset or subclass of $S$.
Let $\preceq'$ be the restriction of $\preceq$ to $T$.
Let $m \in T$.
Then $m$ is the $\preceq$-smallest element of $T$ if and only if $m$ is the $\preceq'$-smallest element of $T$.
Proof
This theorem requires a proof. In particular: The same sort of utterly trivial thing as at Minimal WRT Restriction You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |