src/HOL/Analysis/Path_Connected.thy
changeset 69712 dc85b5b3a532
parent 69661 a03a63b81f44
child 69939 812ce526da33
equal deleted inserted replaced
69711:82a604715919 69712:dc85b5b3a532
  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   }