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Removing Points

The initial work in this direction is found in the paper [95] by Rival, which has since been extended in various directions. We will present the best folklore version of Rival's original result in Theorem 3.2 and a modification in Theorem 3.4. Notice that chain-completeness is not needed in Theorem 3.4 and Scholium 3.5.


Proof: The direction `` tex2html_wrap_inline8128 " follows from part 1 of Theorem 3.1. To prove `` tex2html_wrap_inline8148 " let tex2html_wrap_inline8954 be a comparative retraction and let tex2html_wrap_inline7878 be order-preserving. Then tex2html_wrap_inline8958 has a fixed point p and since r is a comparative retraction f(p) is related to p. Thus by the Abian-Brown-Pelczar Theorem we have that f has a fixed point. \