equal
deleted
inserted
replaced
199 qed |
199 qed |
200 |
200 |
201 lemma%unimportant segment_to_rel_frontier: |
201 lemma%unimportant segment_to_rel_frontier: |
202 fixes x :: "'a::euclidean_space" |
202 fixes x :: "'a::euclidean_space" |
203 assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S" |
203 assumes S: "convex S" "bounded S" and x: "x \<in> rel_interior S" |
204 and y: "y \<in> S" and xy: "~(x = y \<and> S = {x})" |
204 and y: "y \<in> S" and xy: "\<not>(x = y \<and> S = {x})" |
205 obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z" |
205 obtains z where "z \<in> rel_frontier S" "y \<in> closed_segment x z" |
206 "open_segment x z \<subseteq> rel_interior S" |
206 "open_segment x z \<subseteq> rel_interior S" |
207 proof (cases "x=y") |
207 proof (cases "x=y") |
208 case True |
208 case True |
209 with xy have "S \<noteq> {x}" |
209 with xy have "S \<noteq> {x}" |