equal
deleted
inserted
replaced
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" |