src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
changeset 62343 24106dc44def
parent 62217 527488dc8b90
child 62379 340738057c8c
equal deleted inserted replaced
62342:1cf129590be8 62343:24106dc44def
  6688   have 1: "openin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
  6688   have 1: "openin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
  6689     apply (rule open_subset, force)
  6689     apply (rule open_subset, force)
  6690     using \<open>open s\<close>
  6690     using \<open>open s\<close>
  6691     apply (simp add: open_contains_ball Ball_def)
  6691     apply (simp add: open_contains_ball Ball_def)
  6692     apply (erule all_forward)
  6692     apply (erule all_forward)
  6693     using "*" by blast
  6693     using "*" by auto blast+
  6694   have 2: "closedin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
  6694   have 2: "closedin (subtopology euclidean s) (\<Inter>n. {w \<in> s. (deriv ^^ n) f w = 0})"
  6695     using assms
  6695     using assms
  6696     by (auto intro: continuous_closed_in_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
  6696     by (auto intro: continuous_closed_in_preimage_constant holomorphic_on_imp_continuous_on holomorphic_higher_deriv)
  6697   obtain e where "e>0" and e: "ball w e \<subseteq> s" using openE [OF \<open>open s\<close> \<open>w \<in> s\<close>] .
  6697   obtain e where "e>0" and e: "ball w e \<subseteq> s" using openE [OF \<open>open s\<close> \<open>w \<in> s\<close>] .
  6698   then have holfb: "f holomorphic_on ball w e"
  6698   then have holfb: "f holomorphic_on ball w e"