Extension of an order automorphism from a dense subset
= Extension of an order automorphism from a dense subset
An <order automorphism> of an <order-dense subset> $D$ of the real line extends uniquely by $\widetilde\varphi(x)=\sup\{\varphi(d):d\in D,d<x\}$. <Order completeness> supplies the extension, and the lower cuts in $D$ give uniqueness.