equal
deleted
inserted
replaced
2849 { fix x y |
2849 { fix x y |
2850 assume x: "x \<in> interior s" and y: "y \<notin> s" |
2850 assume x: "x \<in> interior s" and y: "y \<notin> s" |
2851 and cc: "connected_component (- frontier s) x y" |
2851 and cc: "connected_component (- frontier s) x y" |
2852 have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}" |
2852 have "connected_component_set (- frontier s) x \<inter> frontier s \<noteq> {}" |
2853 apply (rule connected_Int_frontier, simp) |
2853 apply (rule connected_Int_frontier, simp) |
2854 apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq set_rev_mp x) |
2854 apply (metis IntI cc connected_component_in connected_component_refl empty_iff interiorE mem_Collect_eq rev_subsetD x) |
2855 using y cc |
2855 using y cc |
2856 by blast |
2856 by blast |
2857 then have "bounded (connected_component_set (- frontier s) x)" |
2857 then have "bounded (connected_component_set (- frontier s) x)" |
2858 using connected_component_in by auto |
2858 using connected_component_in by auto |
2859 } |
2859 } |