diff --git a/src/HOL/Analysis/Elementary_Normed_Spaces.thy b/src/HOL/Analysis/Elementary_Normed_Spaces.thy
--- a/src/HOL/Analysis/Elementary_Normed_Spaces.thy
+++ b/src/HOL/Analysis/Elementary_Normed_Spaces.thy
@@ -1,1794 +1,1755 @@
 (*  Author:     L C Paulson, University of Cambridge
     Author:     Amine Chaieb, University of Cambridge
     Author:     Robert Himmelmann, TU Muenchen
     Author:     Brian Huffman, Portland State University
 *)
 
 section \<open>Elementary Normed Vector Spaces\<close>
 
 theory Elementary_Normed_Spaces
   imports
   "HOL-Library.FuncSet"
   Elementary_Metric_Spaces Cartesian_Space
   Connected
 begin
 subsection \<open>Orthogonal Transformation of Balls\<close>
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas Combining Imports\<close>
 
 lemma open_sums:
   fixes T :: "('b::real_normed_vector) set"
   assumes "open S \<or> open T"
   shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
   using assms
 proof
   assume S: "open S"
   show ?thesis
   proof (clarsimp simp: open_dist)
     fix x y
     assume "x \<in> S" "y \<in> T"
     with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
       by (auto simp: open_dist)
     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
       by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
       using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
   qed
 next
   assume T: "open T"
   show ?thesis
   proof (clarsimp simp: open_dist)
     fix x y
     assume "x \<in> S" "y \<in> T"
     with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
       by (auto simp: open_dist)
     then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
       by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
     then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
       using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
   qed
 qed
 
 lemma image_orthogonal_transformation_ball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` ball x r = ball (f x) r"
 proof (intro equalityI subsetI)
   fix y assume "y \<in> f ` ball x r"
   with assms show "y \<in> ball (f x) r"
     by (auto simp: orthogonal_transformation_isometry)
 next
   fix y assume y: "y \<in> ball (f x) r"
   then obtain z where z: "y = f z"
     using assms orthogonal_transformation_surj by blast
   with y assms show "y \<in> f ` ball x r"
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
 lemma image_orthogonal_transformation_cball:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
   assumes "orthogonal_transformation f"
   shows "f ` cball x r = cball (f x) r"
 proof (intro equalityI subsetI)
   fix y assume "y \<in> f ` cball x r"
   with assms show "y \<in> cball (f x) r"
     by (auto simp: orthogonal_transformation_isometry)
 next
   fix y assume y: "y \<in> cball (f x) r"
   then obtain z where z: "y = f z"
     using assms orthogonal_transformation_surj by blast
   with y assms show "y \<in> f ` cball x r"
     by (auto simp: orthogonal_transformation_isometry)
 qed
 
 
 subsection \<open>Support\<close>
 
 definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
-  where "support_on s f = {x\<in>s. f x \<noteq> 0}"
+  where "support_on S f = {x\<in>S. f x \<noteq> 0}"
 
-lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
+lemma in_support_on: "x \<in> support_on S f \<longleftrightarrow> x \<in> S \<and> f x \<noteq> 0"
   by (simp add: support_on_def)
 
 lemma support_on_simps[simp]:
   "support_on {} f = {}"
-  "support_on (insert x s) f =
-    (if f x = 0 then support_on s f else insert x (support_on s f))"
-  "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
-  "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
-  "support_on (s - t) f = support_on s f - support_on t f"
-  "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
+  "support_on (insert x S) f =
+    (if f x = 0 then support_on S f else insert x (support_on S f))"
+  "support_on (S \<union> T) f = support_on S f \<union> support_on T f"
+  "support_on (S \<inter> T) f = support_on S f \<inter> support_on T f"
+  "support_on (S - T) f = support_on S f - support_on T f"
+  "support_on (f ` S) g = f ` (support_on S (g \<circ> f))"
   unfolding support_on_def by auto
 
 lemma support_on_cong:
-  "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
+  "(\<And>x. x \<in> S \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on S f = support_on S g"
   by (auto simp: support_on_def)
 
 lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
   by (auto simp: support_on_def)
 
 lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
   by (auto simp: support_on_def)
 
 lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
   unfolding support_on_def by auto
 
 (* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
 definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
   where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
 
 lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
   unfolding supp_sum_def by auto
 
 lemma supp_sum_insert[simp]:
   "finite (support_on S f) \<Longrightarrow>
     supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
   by (simp add: supp_sum_def in_support_on insert_absorb)
 
 lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
   by (cases "r = 0")
      (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
 
 
 subsection \<open>Intervals\<close>
 
 lemma image_affinity_interval:
   fixes c :: "'a::ordered_real_vector"
   shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = 
            (if {a..b}={} then {}
             else if 0 \<le> m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
             else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
          (is "?lhs = ?rhs")
 proof (cases "m=0")
   case True
   then show ?thesis
     by force
 next
   case False
   show ?thesis
   proof
     show "?lhs \<subseteq> ?rhs"
       by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
     show "?rhs \<subseteq> ?lhs"
     proof (clarsimp, intro conjI impI subsetI)
       show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
             \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
-        apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
-        using False apply (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)
-        done
+        using False
+        by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
+           (auto simp: pos_le_divideR_eq pos_divideR_le_eq le_diff_eq diff_le_eq)
       show "\<lbrakk>\<not> 0 \<le> m; a \<le> b;  x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
             \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
-        apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
-         apply (auto simp add: neg_le_divideR_eq neg_divideR_le_eq not_le le_diff_eq diff_le_eq)
-        done
+        by (rule_tac x="inverse m *\<^sub>R (x-c)" in image_eqI)
+           (auto simp add: neg_le_divideR_eq neg_divideR_le_eq le_diff_eq diff_le_eq)
     qed
   qed
 qed
 
 subsection \<open>Limit Points\<close>
 
 lemma islimpt_ball:
   fixes x y :: "'a::{real_normed_vector,perfect_space}"
   shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   show ?rhs if ?lhs
   proof
     {
       assume "e \<le> 0"
       then have *: "ball x e = {}"
         using ball_eq_empty[of x e] by auto
       have False using \<open>?lhs\<close>
         unfolding * using islimpt_EMPTY[of y] by auto
     }
     then show "e > 0" by (metis not_less)
     show "y \<in> cball x e"
       using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
         ball_subset_cball[of x e] \<open>?lhs\<close>
       unfolding closed_limpt by auto
   qed
   show ?lhs if ?rhs
   proof -
     from that have "e > 0" by auto
     {
       fix d :: real
       assume "d > 0"
       have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
       proof (cases "d \<le> dist x y")
         case True
-        then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+        then show ?thesis
         proof (cases "x = y")
           case True
           then have False
             using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
-          then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
+          then show ?thesis
             by auto
         next
           case False
           have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
             norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
             unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
             by auto
           also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
             using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
             unfolding scaleR_minus_left scaleR_one
             by (auto simp: norm_minus_commute)
           also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
             unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
             unfolding distrib_right using \<open>x\<noteq>y\<close>  by auto
           also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>
             by (auto simp: dist_norm)
           finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>
             by auto
           moreover
           have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
             using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
             by (auto simp: dist_commute)
           moreover
           have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
-            unfolding dist_norm
-            apply simp
-            unfolding norm_minus_cancel
-            using \<open>d > 0\<close> \<open>x\<noteq>y\<close>[unfolded dist_nz] dist_commute[of x y]
-            unfolding dist_norm
-            apply auto
-            done
-          ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
-            apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
-            apply auto
-            done
+            using \<open>0 < d\<close> by (fastforce simp: dist_norm)
+          ultimately show ?thesis
+            by (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
         qed
       next
         case False
         then have "d > dist x y" by auto
         show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
         proof (cases "x = y")
           case True
-          obtain z where **: "z \<noteq> y" "dist z y < min e d"
+          obtain z where z: "z \<noteq> y" "dist z y < min e d"
             using perfect_choose_dist[of "min e d" y]
             using \<open>d > 0\<close> \<open>e>0\<close> by auto
-          show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
-            unfolding \<open>x = y\<close>
-            using \<open>z \<noteq> y\<close> **
-            apply (rule_tac x=z in bexI)
-            apply (auto simp: dist_commute)
-            done
+          show ?thesis
+            by (metis True z dist_commute mem_ball min_less_iff_conj)
         next
           case False
-          then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
-            using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close>
-            apply (rule_tac x=x in bexI, auto)
-            done
+          then show ?thesis
+            using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close> by force
         qed
       qed
     }
     then show ?thesis
       unfolding mem_cball islimpt_approachable mem_ball by auto
   qed
 qed
 
 lemma closure_ball_lemma:
   fixes x y :: "'a::real_normed_vector"
   assumes "x \<noteq> y"
   shows "y islimpt ball x (dist x y)"
 proof (rule islimptI)
   fix T
   assume "y \<in> T" "open T"
   then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
     unfolding open_dist by fast
-  (* choose point between x and y, within distance r of y. *)
+  \<comment>\<open>choose point between @{term x} and @{term y}, within distance @{term r} of @{term y}.\<close>
   define k where "k = min 1 (r / (2 * dist x y))"
   define z where "z = y + scaleR k (x - y)"
   have z_def2: "z = x + scaleR (1 - k) (y - x)"
     unfolding z_def by (simp add: algebra_simps)
   have "dist z y < r"
     unfolding z_def k_def using \<open>0 < r\<close>
     by (simp add: dist_norm min_def)
   then have "z \<in> T"
     using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
   have "dist x z < dist x y"
-    unfolding z_def2 dist_norm
-    apply (simp add: norm_minus_commute)
-    apply (simp only: dist_norm [symmetric])
-    apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
-    apply (rule mult_strict_right_mono)
-    apply (simp add: k_def \<open>0 < r\<close> \<open>x \<noteq> y\<close>)
-    apply (simp add: \<open>x \<noteq> y\<close>)
-    done
+    using \<open>0 < r\<close> assms by (simp add: z_def2 k_def dist_norm norm_minus_commute) 
   then have "z \<in> ball x (dist x y)"
     by simp
   have "z \<noteq> y"
     unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
     by (simp add: min_def)
   show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
     using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
     by fast
 qed
 
 
 subsection \<open>Balls and Spheres in Normed Spaces\<close>
 
 lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
   for x :: "'a::real_normed_vector"
-  by (simp add: dist_norm)
+  by simp
 
 lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
   for x :: "'a::real_normed_vector"
-  by (simp add: dist_norm)
+  by simp
 
 lemma closure_ball [simp]:
   fixes x :: "'a::real_normed_vector"
-  shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
-  apply (rule equalityI)
-  apply (rule closure_minimal)
-  apply (rule ball_subset_cball)
-  apply (rule closed_cball)
-  apply (rule subsetI, rename_tac y)
-  apply (simp add: le_less [where 'a=real])
-  apply (erule disjE)
-  apply (rule subsetD [OF closure_subset], simp)
-  apply (simp add: closure_def, clarify)
-  apply (rule closure_ball_lemma)
-  apply simp
-  done
+  assumes "0 < e"
+  shows "closure (ball x e) = cball x e"
+proof
+  show "closure (ball x e) \<subseteq> cball x e"
+    using closed_cball closure_minimal by blast
+  have "\<And>y. dist x y < e \<or> dist x y = e \<Longrightarrow> y \<in> closure (ball x e)"
+    by (metis Un_iff assms closure_ball_lemma closure_def dist_eq_0_iff mem_Collect_eq mem_ball)
+  then show "cball x e \<subseteq> closure (ball x e)"
+    by force
+qed
 
 lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
   for x :: "'a::real_normed_vector"
-  by (simp add: dist_norm)
+  by simp
 
 (* In a trivial vector space, this fails for e = 0. *)
 lemma interior_cball [simp]:
   fixes x :: "'a::{real_normed_vector, perfect_space}"
   shows "interior (cball x e) = ball x e"
 proof (cases "e \<ge> 0")
   case False note cs = this
   from cs have null: "ball x e = {}"
     using ball_empty[of e x] by auto
   moreover
   have "cball x e = {}"
   proof (rule equals0I)
     fix y
     assume "y \<in> cball x e"
     then show False
       by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball
           subset_antisym subset_ball)
   qed
   then have "interior (cball x e) = {}"
     using interior_empty by auto
   ultimately show ?thesis by blast
 next
   case True note cs = this
   have "ball x e \<subseteq> cball x e"
     using ball_subset_cball by auto
   moreover
   {
     fix S y
     assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
     then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
       unfolding open_dist by blast
     then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
       using perfect_choose_dist [of d] by auto
     have "xa \<in> S"
       using d[THEN spec[where x = xa]]
       using xa by (auto simp: dist_commute)
     then have xa_cball: "xa \<in> cball x e"
       using as(1) by auto
     then have "y \<in> ball x e"
     proof (cases "x = y")
       case True
       then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
       then show "y \<in> ball x e"
         using \<open>x = y \<close> by simp
     next
       case False
       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
         unfolding dist_norm
         using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
       then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
         using d as(1)[unfolded subset_eq] by blast
       have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto
       hence **:"d / (2 * norm (y - x)) > 0"
         unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
       have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
         norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
         by (auto simp: dist_norm algebra_simps)
       also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
         by (auto simp: algebra_simps)
       also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
         using ** by auto
       also have "\<dots> = (dist y x) + d/2"
         using ** by (auto simp: distrib_right dist_norm)
       finally have "e \<ge> dist x y +d/2"
         using *[unfolded mem_cball] by (auto simp: dist_commute)
       then show "y \<in> ball x e"
         unfolding mem_ball using \<open>d>0\<close> by auto
     qed
   }
   then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
     by auto
   ultimately show ?thesis
     using interior_unique[of "ball x e" "cball x e"]
     using open_ball[of x e]
     by auto
 qed
 
 lemma frontier_ball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"
   by (force simp: frontier_def)
 
 lemma frontier_cball [simp]:
   fixes a :: "'a::{real_normed_vector, perfect_space}"
   shows "frontier (cball a e) = sphere a e"
   by (force simp: frontier_def)
 
 corollary compact_sphere [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "compact (sphere a r)"
 using compact_frontier [of "cball a r"] by simp
 
 corollary bounded_sphere [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "bounded (sphere a r)"
 by (simp add: compact_imp_bounded)
 
 corollary closed_sphere  [simp]:
   fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
   shows "closed (sphere a r)"
 by (simp add: compact_imp_closed)
 
 lemma image_add_ball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "(+) b ` ball a r = ball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
+proof -
+  { fix x :: 'a
+    assume "dist (a + b) x < r"
+    moreover
+    have "b + (x - b) = x"
+      by simp
+    ultimately have "x \<in> (+) b ` ball a r"
+      by (metis add.commute dist_add_cancel image_eqI mem_ball) }
+  then show ?thesis
+    by (auto simp: add.commute)
+qed
 
 lemma image_add_cball [simp]:
   fixes a :: "'a::real_normed_vector"
   shows "(+) b ` cball a r = cball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
+proof -
+  have "\<And>x. dist (a + b) x \<le> r \<Longrightarrow> \<exists>y\<in>cball a r. x = b + y"
+    by (metis (no_types) add.commute diff_add_cancel dist_add_cancel2 mem_cball)
+  then show ?thesis
+    by (force simp: add.commute)
+qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Various Lemmas on Normed Algebras\<close>
 
 
 lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
 
 lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
   by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
 
 lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
   unfolding Nats_def by (rule closed_of_nat_image)
 
 lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
   unfolding Ints_def by (rule closed_of_int_image)
 
 lemma closed_subset_Ints:
   fixes A :: "'a :: real_normed_algebra_1 set"
   assumes "A \<subseteq> \<int>"
   shows   "closed A"
 proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
   case (1 x y)
   with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
   with \<open>dist y x < 1\<close> show "y = x"
     by (auto elim!: Ints_cases simp: dist_of_int)
 qed
 
 subsection \<open>Filters\<close>
 
 definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"  (infixr "indirection" 70)
   where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
 
 
 subsection \<open>Trivial Limits\<close>
 
 lemma trivial_limit_at_infinity:
   "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
-  unfolding trivial_limit_def eventually_at_infinity
-  apply clarsimp
-  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
-   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
-  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
-  apply (drule_tac x=UNIV in spec, simp)
-  done
+proof -
+  obtain x::'a where "x \<noteq> 0"
+    by (meson perfect_choose_dist zero_less_one)
+  then have "b \<le> norm ((b / norm x) *\<^sub>R x)" for b
+    by simp
+  then show ?thesis
+    unfolding trivial_limit_def eventually_at_infinity
+    by blast
+qed
 
 lemma at_within_ball_bot_iff:
   fixes x y :: "'a::{real_normed_vector,perfect_space}"
   shows "at x within ball y r = bot \<longleftrightarrow> (r=0 \<or> x \<notin> cball y r)"
-  unfolding trivial_limit_within 
-  apply (auto simp add:trivial_limit_within islimpt_ball )
-  by (metis le_less_trans less_eq_real_def zero_le_dist)
+  unfolding trivial_limit_within
+  by (metis (no_types) cball_empty equals0D islimpt_ball less_linear) 
 
 
 subsection \<open>Limits\<close>
 
 proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
   by (auto simp: tendsto_iff eventually_at_infinity)
 
 corollary Lim_at_infinityI [intro?]:
   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
   shows "(f \<longlongrightarrow> l) at_infinity"
-  apply (simp add: Lim_at_infinity, clarify)
-  apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
-  done
+proof -
+  have "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l < e"
+    by (meson assms dense le_less_trans)
+  then show ?thesis
+    using Lim_at_infinity by blast
+qed
 
 lemma Lim_transform_within_set_eq:
   fixes a :: "'a::metric_space" and l :: "'b::metric_space"
   shows "eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)
          \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within T))"
   by (force intro: Lim_transform_within_set elim: eventually_mono)
 
 lemma Lim_null:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
   by (simp add: Lim dist_norm)
 
 lemma Lim_null_comparison:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
   shows "(f \<longlongrightarrow> 0) net"
   using assms(2)
 proof (rule metric_tendsto_imp_tendsto)
   show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
     using assms(1) by (rule eventually_mono) (simp add: dist_norm)
 qed
 
 lemma Lim_transform_bound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
     and g :: "'a \<Rightarrow> 'c::real_normed_vector"
   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
     and "(g \<longlongrightarrow> 0) net"
   shows "(f \<longlongrightarrow> 0) net"
   using assms(1) tendsto_norm_zero [OF assms(2)]
   by (rule Lim_null_comparison)
 
 lemma lim_null_mult_right_bounded:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
   assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
     shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
 proof -
-  have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
-    by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
   have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
-    apply (rule Lim_null_comparison [OF _ *])
-    apply (simp add: eventually_mono [OF g] mult_left_mono)
-    done
+  proof (rule Lim_null_comparison)
+    show "\<forall>\<^sub>F x in F. norm (norm (f x) * norm (g x)) \<le> norm (f x) * B"
+      by (simp add: eventually_mono [OF g] mult_left_mono)
+    show "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
+      by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
+  qed
   then show ?thesis
     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
 qed
 
 lemma lim_null_mult_left_bounded:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
   assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
     shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
 proof -
-  have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
-    by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
   have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
-    apply (rule Lim_null_comparison [OF _ *])
-    apply (simp add: eventually_mono [OF g] mult_right_mono)
-    done
+  proof (rule Lim_null_comparison)
+    show "\<forall>\<^sub>F x in F. norm (norm (g x) * norm (f x)) \<le> B * norm (f x)"
+      by (simp add: eventually_mono [OF g] mult_right_mono)
+    show "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
+      by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
+  qed
   then show ?thesis
     by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
 qed
 
 lemma lim_null_scaleR_bounded:
   assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
     shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
 proof
   fix \<epsilon>::real
   assume "0 < \<epsilon>"
   then have B: "0 < \<epsilon> / (abs B + 1)" by simp
   have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
   proof -
     have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
       by (simp add: mult_left_mono g)
     also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
       by (simp add: mult_left_mono)
     also have "\<dots> < \<epsilon>"
       by (rule f)
     finally show ?thesis .
   qed
-  show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
-    apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
-    apply (auto simp: \<open>0 < \<epsilon>\<close> field_split_simps * split: if_split_asm)
-    done
+  have "\<And>x. \<lbrakk>\<bar>f x\<bar> < \<epsilon> / (\<bar>B\<bar> + 1); norm (g x) \<le> B\<rbrakk> \<Longrightarrow> \<bar>f x\<bar> * norm (g x) < \<epsilon>"
+    by (simp add: "*" pos_less_divide_eq)
+  then show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
+    using \<open>0 < \<epsilon>\<close> by (auto intro: eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB]])
 qed
 
 lemma Lim_norm_ubound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
   shows "norm(l) \<le> e"
   using assms by (fast intro: tendsto_le tendsto_intros)
 
 lemma Lim_norm_lbound:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "\<not> trivial_limit net"
     and "(f \<longlongrightarrow> l) net"
     and "eventually (\<lambda>x. e \<le> norm (f x)) net"
   shows "e \<le> norm l"
   using assms by (fast intro: tendsto_le tendsto_intros)
 
 text\<open>Limit under bilinear function\<close>
 
 lemma Lim_bilinear:
   assumes "(f \<longlongrightarrow> l) net"
     and "(g \<longlongrightarrow> m) net"
     and "bounded_bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
   using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
   by (rule bounded_bilinear.tendsto)
 
 lemma Lim_at_zero:
   fixes a :: "'a::real_normed_vector"
     and l :: "'b::topological_space"
   shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
   using LIM_offset_zero LIM_offset_zero_cancel ..
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Limit Point of Filter\<close>
 
 lemma netlimit_at_vector:
   fixes a :: "'a::real_normed_vector"
   shows "netlimit (at a) = a"
 proof (cases "\<exists>x. x \<noteq> a")
   case True then obtain x where x: "x \<noteq> a" ..
-  have "\<not> trivial_limit (at a)"
-    unfolding trivial_limit_def eventually_at dist_norm
-    apply clarsimp
-    apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
-    apply (simp add: norm_sgn sgn_zero_iff x)
-    done
+  have "\<And>d. 0 < d \<Longrightarrow> \<exists>x. x \<noteq> a \<and> norm (x - a) < d"
+    by (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI) (simp add: norm_sgn sgn_zero_iff x)
+  then have "\<not> trivial_limit (at a)"
+    by (auto simp: trivial_limit_def eventually_at dist_norm)
   then show ?thesis
     by (rule Lim_ident_at [of a UNIV])
 qed simp
 
 subsection \<open>Boundedness\<close>
 
 lemma continuous_on_closure_norm_le:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "continuous_on (closure s) f"
     and "\<forall>y \<in> s. norm(f y) \<le> b"
     and "x \<in> (closure s)"
   shows "norm (f x) \<le> b"
 proof -
   have *: "f ` s \<subseteq> cball 0 b"
     using assms(2)[unfolded mem_cball_0[symmetric]] by auto
   show ?thesis
     by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
 qed
 
 lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
   unfolding bounded_iff 
   by (meson less_imp_le not_le order_trans zero_less_one)
 
 lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
-  apply (simp add: bounded_pos)
-  apply (safe; rule_tac x="b+1" in exI; force)
-  done
+  by (metis bounded_pos le_less_trans less_imp_le linordered_field_no_ub)
 
 lemma Bseq_eq_bounded:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   shows "Bseq f \<longleftrightarrow> bounded (range f)"
   unfolding Bseq_def bounded_pos by auto
 
 lemma bounded_linear_image:
   assumes "bounded S"
     and "bounded_linear f"
   shows "bounded (f ` S)"
 proof -
   from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
     unfolding bounded_pos by auto
   from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
     using bounded_linear.pos_bounded by (auto simp: ac_simps)
   show ?thesis
     unfolding bounded_pos
   proof (intro exI, safe)
     show "norm (f x) \<le> B * b" if "x \<in> S" for x
       by (meson B b less_imp_le mult_left_mono order_trans that)
   qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
 qed
 
 lemma bounded_scaling:
   fixes S :: "'a::real_normed_vector set"
   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
-  apply (rule bounded_linear_image, assumption)
-  apply (rule bounded_linear_scaleR_right)
-  done
+  by (simp add: bounded_linear_image bounded_linear_scaleR_right)
 
 lemma bounded_scaleR_comp:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   assumes "bounded (f ` S)"
   shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
   using bounded_scaling[of "f ` S" r] assms
   by (auto simp: image_image)
 
 lemma bounded_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "bounded S"
   shows "bounded ((\<lambda>x. a + x) ` S)"
 proof -
   from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
     unfolding bounded_pos by auto
   {
     fix x
     assume "x \<in> S"
     then have "norm (a + x) \<le> b + norm a"
       using norm_triangle_ineq[of a x] b by auto
   }
   then show ?thesis
     unfolding bounded_pos
     using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
     by (auto intro!: exI[of _ "b + norm a"])
 qed
 
 lemma bounded_translation_minus:
   fixes S :: "'a::real_normed_vector set"
   shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
 using bounded_translation [of S "-a"] by simp
 
 lemma bounded_uminus [simp]:
   fixes X :: "'a::real_normed_vector set"
   shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
 by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
 
 lemma uminus_bounded_comp [simp]:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
   shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
   using bounded_uminus[of "f ` S"]
   by (auto simp: image_image)
 
 lemma bounded_plus_comp:
   fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
   assumes "bounded (f ` S)"
   assumes "bounded (g ` S)"
   shows "bounded ((\<lambda>x. f x + g x) ` S)"
 proof -
   {
     fix B C
     assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
     then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
       by (auto intro!: norm_triangle_le add_mono)
   } then show ?thesis
     using assms by (fastforce simp: bounded_iff)
 qed
 
 lemma bounded_plus:
   fixes S ::"'a::real_normed_vector set"
   assumes "bounded S" "bounded T"
   shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
   using bounded_plus_comp [of fst "S \<times> T" snd] assms
   by (auto simp: split_def split: if_split_asm)
 
 lemma bounded_minus_comp:
   "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
   for f g::"'a \<Rightarrow> 'b::real_normed_vector"
   using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
   by auto
 
 lemma bounded_minus:
   fixes S ::"'a::real_normed_vector set"
   assumes "bounded S" "bounded T"
   shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
   using bounded_minus_comp [of fst "S \<times> T" snd] assms
   by (auto simp: split_def split: if_split_asm)
 
 lemma not_bounded_UNIV[simp]:
   "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
 proof (auto simp: bounded_pos not_le)
   obtain x :: 'a where "x \<noteq> 0"
     using perfect_choose_dist [OF zero_less_one] by fast
   fix b :: real
   assume b: "b >0"
   have b1: "b +1 \<ge> 0"
     using b by simp
   with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
     by (simp add: norm_sgn)
   then show "\<exists>x::'a. b < norm x" ..
 qed
 
 corollary cobounded_imp_unbounded:
     fixes S :: "'a::{real_normed_vector, perfect_space} set"
     shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
   using bounded_Un [of S "-S"]  by (simp)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Relations among convergence and absolute convergence for power series\<close>
 
 lemma summable_imp_bounded:
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   shows "summable f \<Longrightarrow> bounded (range f)"
 by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
 
 lemma summable_imp_sums_bounded:
    "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
 by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
 
 lemma power_series_conv_imp_absconv_weak:
   fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a
   assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"
     shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
 proof -
   obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
     using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
-  then have *: "summable (\<lambda>n. norm (a n) * norm w ^ n)"
-    by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
   show ?thesis
-    apply (rule series_comparison_complex [of "(\<lambda>n. of_real(norm(a n) * norm w ^ n))"])
-    apply (simp only: summable_complex_of_real *)
-    apply (auto simp: norm_mult norm_power)
-    done
+  proof (rule series_comparison_complex)
+    have "\<And>n. norm (a n) * norm z ^ n \<le> M"
+      by (metis (no_types) M norm_mult norm_power)
+    then show "summable (\<lambda>n. complex_of_real (norm (a n) * norm w ^ n))"
+      using Abel_lemma no norm_ge_zero summable_of_real by blast
+  qed (auto simp: norm_mult norm_power)
 qed
 
 
 subsection \<open>Normed spaces with the Heine-Borel property\<close>
 
 lemma not_compact_UNIV[simp]:
   fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
   shows "\<not> compact (UNIV::'a set)"
     by (simp add: compact_eq_bounded_closed)
 
 lemma not_compact_space_euclideanreal [simp]: "\<not> compact_space euclideanreal"
   by (simp add: compact_space_def)
 
 text\<open>Representing sets as the union of a chain of compact sets.\<close>
 lemma closed_Union_compact_subsets:
   fixes S :: "'a::{heine_borel,real_normed_vector} set"
   assumes "closed S"
   obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
                   "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
 proof
   show "compact (S \<inter> cball 0 (of_nat n))" for n
     using assms compact_eq_bounded_closed by auto
 next
   show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
     by (auto simp: real_arch_simple)
 next
   fix K :: "'a set"
   assume "compact K" "K \<subseteq> S"
   then obtain N where "K \<subseteq> cball 0 N"
     by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
   then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
     by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
 qed auto
 
 subsection \<open>Intersecting chains of compact sets and the Baire property\<close>
 
 proposition bounded_closed_chain:
   fixes \<F> :: "'a::heine_borel set set"
   assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"
       and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     shows "\<Inter>\<F> \<noteq> {}"
 proof -
   have "B \<inter> \<Inter>\<F> \<noteq> {}"
   proof (rule compact_imp_fip)
     show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
       by (simp_all add: assms compact_eq_bounded_closed)
     show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>
     proof (induction \<G> rule: finite_induct)
       case empty
       with assms show ?case by force
     next
       case (insert U \<G>)
       then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto
       then consider "B \<subseteq> U" | "U \<subseteq> B"
           using \<open>B \<in> \<F>\<close> chain by blast
         then show ?case
         proof cases
           case 1
           then show ?thesis
             using Int_left_commute ne by auto
         next
           case 2
           have "U \<noteq> {}"
             using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
           moreover
           have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
           proof -
             have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"
               by (metis chain contra_subsetD insert.prems insert_subset that)
             then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"
               by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
             moreover obtain x where "x \<in> \<Inter>\<G>"
               by (metis Int_emptyI ne)
             ultimately show ?thesis
               by (metis Inf_lower subset_eq that)
           qed
           with 2 show ?thesis
             by blast
         qed
       qed
   qed
   then show ?thesis by blast
 qed
 
 corollary compact_chain:
   fixes \<F> :: "'a::heine_borel set set"
   assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"
           "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     shows "\<Inter> \<F> \<noteq> {}"
 proof (cases "\<F> = {}")
   case True
   then show ?thesis by auto
 next
   case False
   show ?thesis
     by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
 qed
 
 lemma compact_nest:
   fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
   assumes F: "\<And>n. compact(F n)" "\<And>n. F n \<noteq> {}" and mono: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
   shows "\<Inter>(range F) \<noteq> {}"
 proof -
   have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
     by (metis mono image_iff le_cases)
   show ?thesis
-    apply (rule compact_chain [OF _ _ *])
-    using F apply (blast intro: dest: *)+
-    done
+    using F by (intro compact_chain [OF _ _ *]; blast dest: *)
 qed
 
 text\<open>The Baire property of dense sets\<close>
 theorem Baire:
   fixes S::"'a::{real_normed_vector,heine_borel} set"
   assumes "closed S" "countable \<G>"
       and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (top_of_set S) T \<and> S \<subseteq> closure T"
  shows "S \<subseteq> closure(\<Inter>\<G>)"
 proof (cases "\<G> = {}")
   case True
   then show ?thesis
     using closure_subset by auto
 next
   let ?g = "from_nat_into \<G>"
   case False
   then have gin: "?g n \<in> \<G>" for n
     by (simp add: from_nat_into)
   show ?thesis
   proof (clarsimp simp: closure_approachable)
     fix x and e::real
     assume "x \<in> S" "0 < e"
     obtain TF where opeF: "\<And>n. openin (top_of_set S) (TF n)"
                and ne: "\<And>n. TF n \<noteq> {}"
                and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"
                and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"
                and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
     proof -
       have *: "\<exists>Y. (openin (top_of_set S) Y \<and> Y \<noteq> {} \<and>
                    S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
         if opeU: "openin (top_of_set S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
       proof -
         obtain T where T: "open T" "U = T \<inter> S"
           using \<open>openin (top_of_set S) U\<close> by (auto simp: openin_subtopology)
         with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
           using gin ope by fastforce
         then have "T \<inter> ?g n \<noteq> {}"
           using \<open>open T\<close> open_Int_closure_eq_empty by blast
         then obtain y where "y \<in> U" "y \<in> ?g n"
           using T ope [of "?g n", OF gin] by (blast dest:  openin_imp_subset)
         moreover have "openin (top_of_set S) (U \<inter> ?g n)"
           using gin ope opeU by blast
         ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"
           by (force simp: openin_contains_ball)
         show ?thesis
         proof (intro exI conjI)
           show "openin (top_of_set S) (S \<inter> ball y (d/2))"
             by (simp add: openin_open_Int)
           show "S \<inter> ball y (d/2) \<noteq> {}"
             using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
           have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"
             using closure_mono by blast
           also have "... \<subseteq> ?g n"
             using \<open>d > 0\<close> d by force
           finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .
           have "closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> ball y d"
           proof -
             have "closure (ball y (d/2)) \<subseteq> ball y d"
               using \<open>d > 0\<close> by auto
             then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"
               by (meson closure_mono inf.cobounded2 subset_trans)
             then show ?thesis
               by (simp add: \<open>closed S\<close> closure_minimal)
           qed
           also have "...  \<subseteq> ball x e"
             using cloU closure_subset d by blast
           finally show "closure (S \<inter> ball y (d/2)) \<subseteq> ball x e" .
           show "S \<inter> ball y (d/2) \<subseteq> U"
             using ball_divide_subset_numeral d by blast
         qed
       qed
       let ?\<Phi> = "\<lambda>n X. openin (top_of_set S) X \<and> X \<noteq> {} \<and>
                       S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
-      have "closure (S \<inter> ball x (e / 2)) \<subseteq> closure(ball x (e/2))"
+      have "closure (S \<inter> ball x (e/2)) \<subseteq> closure(ball x (e/2))"
         by (simp add: closure_mono)
       also have "...  \<subseteq> ball x e"
         using \<open>e > 0\<close> by auto
-      finally have "closure (S \<inter> ball x (e / 2)) \<subseteq> ball x e" .
-      moreover have"openin (top_of_set S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"
+      finally have "closure (S \<inter> ball x (e/2)) \<subseteq> ball x e" .
+      moreover have"openin (top_of_set S) (S \<inter> ball x (e/2))" "S \<inter> ball x (e/2) \<noteq> {}"
         using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
-      ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e / 2)"
+      ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e/2)"
             using * [of "S \<inter> ball x (e/2)" 0] by metis
       show thesis
-      proof (rule exE [OF dependent_nat_choice [of ?\<Phi> "\<lambda>n X Y. Y \<subseteq> X"]])
+      proof (rule exE [OF dependent_nat_choice])
         show "\<exists>x. ?\<Phi> 0 x"
           using Y by auto
         show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X" if "?\<Phi> n X" for X n
           using that by (blast intro: *)
       qed (use that in metis)
     qed
     have "(\<Inter>n. S \<inter> closure (TF n)) \<noteq> {}"
     proof (rule compact_nest)
       show "\<And>n. compact (S \<inter> closure (TF n))"
         by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
       show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"
         by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)
       show "\<And>m n. m \<le> n \<Longrightarrow> S \<inter> closure (TF n) \<subseteq> S \<inter> closure (TF m)"
         by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
     qed
     moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>. dist y x < e}"
     proof (clarsimp, intro conjI)
       fix y
       assume "y \<in> S" and y: "\<forall>n. y \<in> closure (TF n)"
       then show "\<forall>T\<in>\<G>. y \<in> T"
         by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] subsetD subg)
       show "dist y x < e"
         by (metis y dist_commute mem_ball subball subsetCE)
     qed
     ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"
       by auto
   qed
 qed
 
 
 subsection \<open>Continuity\<close>
 
 subsubsection\<^marker>\<open>tag unimportant\<close> \<open>Structural rules for uniform continuity\<close>
 
 lemma (in bounded_linear) uniformly_continuous_on[continuous_intros]:
   fixes g :: "_::metric_space \<Rightarrow> _"
   assumes "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
   using assms unfolding uniformly_continuous_on_sequentially
   unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
   by (auto intro: tendsto_zero)
 
 lemma uniformly_continuous_on_dist[continuous_intros]:
   fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
 proof -
   {
     fix a b c d :: 'b
     have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
       using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
       using dist_triangle3 [of c d a] dist_triangle [of a d b]
       by arith
   } note le = this
   {
     fix x y
     assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
     assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
     have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
       by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
         simp add: le)
   }
   then show ?thesis
     using assms unfolding uniformly_continuous_on_sequentially
     unfolding dist_real_def by simp
 qed
 
 lemma uniformly_continuous_on_cmul_right [continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
   using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .
 
 lemma uniformly_continuous_on_cmul_left[continuous_intros]:
   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
   assumes "uniformly_continuous_on s f"
     shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
 by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)
 
 lemma uniformly_continuous_on_norm[continuous_intros]:
   fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
   assumes "uniformly_continuous_on s f"
   shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
   unfolding norm_conv_dist using assms
   by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
 
 lemma uniformly_continuous_on_cmul[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
   shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
   using bounded_linear_scaleR_right assms
   by (rule bounded_linear.uniformly_continuous_on)
 
 lemma dist_minus:
   fixes x y :: "'a::real_normed_vector"
   shows "dist (- x) (- y) = dist x y"
   unfolding dist_norm minus_diff_minus norm_minus_cancel ..
 
 lemma uniformly_continuous_on_minus[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
   unfolding uniformly_continuous_on_def dist_minus .
 
 lemma uniformly_continuous_on_add[continuous_intros]:
   fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
   using assms
   unfolding uniformly_continuous_on_sequentially
   unfolding dist_norm tendsto_norm_zero_iff add_diff_add
   by (auto intro: tendsto_add_zero)
 
 lemma uniformly_continuous_on_diff[continuous_intros]:
   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
   assumes "uniformly_continuous_on s f"
     and "uniformly_continuous_on s g"
   shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
   using assms uniformly_continuous_on_add [of s f "- g"]
     by (simp add: fun_Compl_def uniformly_continuous_on_minus)
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Arithmetic Preserves Topological Properties\<close>
 
 lemma open_scaling[intro]:
   fixes s :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
     and "open s"
   shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
 proof -
   {
     fix x
     assume "x \<in> s"
     then obtain e where "e>0"
       and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
       by auto
     have "e * \<bar>c\<bar> > 0"
       using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
     moreover
     {
       fix y
       assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
+      then have "norm (c *\<^sub>R ((1 / c) *\<^sub>R y - x)) < e * norm c"
+        by (simp add: \<open>c \<noteq> 0\<close> dist_norm scale_right_diff_distrib)
       then have "norm ((1 / c) *\<^sub>R y - x) < e"
-        unfolding dist_norm
-        using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
-          assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
+        by (simp add: \<open>c \<noteq> 0\<close>)
       then have "y \<in> (*\<^sub>R) c ` s"
         using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "(*\<^sub>R) c"]
-        using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
-        using assms(1)
-        unfolding dist_norm scaleR_scaleR
-        by auto
+        by (simp add: \<open>c \<noteq> 0\<close> dist_norm e)
     }
     ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> (*\<^sub>R) c ` s"
-      apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
-      done
+      by (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
   }
   then show ?thesis unfolding open_dist by auto
 qed
 
 lemma minus_image_eq_vimage:
   fixes A :: "'a::ab_group_add set"
   shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
   by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
 
 lemma open_negations:
   fixes S :: "'a::real_normed_vector set"
   shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
   using open_scaling [of "- 1" S] by simp
 
 lemma open_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open((\<lambda>x. a + x) ` S)"
 proof -
   {
     fix x
     have "continuous (at x) (\<lambda>x. x - a)"
       by (intro continuous_diff continuous_ident continuous_const)
   }
   moreover have "{x. x - a \<in> S} = (+) a ` S"
     by force
   ultimately show ?thesis
     by (metis assms continuous_open_vimage vimage_def)
 qed
 
 lemma open_translation_subtract:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open ((\<lambda>x. x - a) ` S)" 
   using assms open_translation [of S "- a"] by (simp cong: image_cong_simp)
 
 lemma open_neg_translation:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"
   shows "open((\<lambda>x. a - x) ` S)"
   using open_translation[OF open_negations[OF assms], of a]
   by (auto simp: image_image)
 
 lemma open_affinity:
   fixes S :: "'a::real_normed_vector set"
   assumes "open S"  "c \<noteq> 0"
   shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
   have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
     unfolding o_def ..
   have "(+) a ` (*\<^sub>R) c ` S = ((+) a \<circ> (*\<^sub>R) c) ` S"
     by auto
   then show ?thesis
     using assms open_translation[of "(*\<^sub>R) c ` S" a]
     unfolding *
     by auto
 qed
 
 lemma interior_translation:
   "interior ((+) a ` S) = (+) a ` (interior S)" for S :: "'a::real_normed_vector set"
 proof (rule set_eqI, rule)
   fix x
   assume "x \<in> interior ((+) a ` S)"
   then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) a ` S"
     unfolding mem_interior by auto
   then have "ball (x - a) e \<subseteq> S"
     unfolding subset_eq Ball_def mem_ball dist_norm
     by (auto simp: diff_diff_eq)
   then show "x \<in> (+) a ` interior S"
     unfolding image_iff
-    apply (rule_tac x="x - a" in bexI)
-    unfolding mem_interior
-    using \<open>e > 0\<close>
-    apply auto
-    done
+    by (metis \<open>0 < e\<close> add.commute diff_add_cancel mem_interior)
 next
   fix x
   assume "x \<in> (+) a ` interior S"
   then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
     unfolding image_iff Bex_def mem_interior by auto
   {
     fix z
     have *: "a + y - z = y + a - z" by auto
     assume "z \<in> ball x e"
     then have "z - a \<in> S"
       using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
       unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
       by auto
     then have "z \<in> (+) a ` S"
       unfolding image_iff by (auto intro!: bexI[where x="z - a"])
   }
   then have "ball x e \<subseteq> (+) a ` S"
     unfolding subset_eq by auto
   then show "x \<in> interior ((+) a ` S)"
     unfolding mem_interior using \<open>e > 0\<close> by auto
 qed
 
 lemma interior_translation_subtract:
   "interior ((\<lambda>x. x - a) ` S) = (\<lambda>x. x - a) ` interior S" for S :: "'a::real_normed_vector set"
   using interior_translation [of "- a"] by (simp cong: image_cong_simp)
 
 
 lemma compact_scaling:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
 proof -
   let ?f = "\<lambda>x. scaleR c x"
   have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
   show ?thesis
     using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
     using linear_continuous_at[OF *] assms
     by auto
 qed
 
 lemma compact_negations:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. - x) ` s)"
   using compact_scaling [OF assms, of "- 1"] by auto
 
 lemma compact_sums:
   fixes s t :: "'a::real_normed_vector set"
   assumes "compact s"
     and "compact t"
   shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
 proof -
   have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
-    apply auto
-    unfolding image_iff
-    apply (rule_tac x="(xa, y)" in bexI)
-    apply auto
-    done
+    by (fastforce simp: image_iff)
   have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
     unfolding continuous_on by (rule ballI) (intro tendsto_intros)
   then show ?thesis
     unfolding * using compact_continuous_image compact_Times [OF assms] by auto
 qed
 
 lemma compact_differences:
   fixes s t :: "'a::real_normed_vector set"
   assumes "compact s"
     and "compact t"
   shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
 proof-
-  have "{x - y | x y. x\<in>s \<and> y \<in> t} =  {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
-    apply auto
-    apply (rule_tac x= xa in exI, auto)
-    done
+  have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
+    using diff_conv_add_uminus by force
   then show ?thesis
     using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
 qed
 
 lemma compact_translation:
   "compact ((+) a ` s)" if "compact s" for s :: "'a::real_normed_vector set"
 proof -
   have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
     by auto
   then show ?thesis
     using compact_sums [OF that compact_sing [of a]] by auto
 qed
 
 lemma compact_translation_subtract:
   "compact ((\<lambda>x. x - a) ` s)" if "compact s" for s :: "'a::real_normed_vector set"
   using that compact_translation [of s "- a"] by (simp cong: image_cong_simp)
 
 lemma compact_affinity:
   fixes s :: "'a::real_normed_vector set"
   assumes "compact s"
   shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
 proof -
   have "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
     by auto
   then show ?thesis
     using compact_translation[OF compact_scaling[OF assms], of a c] by auto
 qed
 
 lemma closed_scaling:
   fixes S :: "'a::real_normed_vector set"
   assumes "closed S"
   shows "closed ((\<lambda>x. c *\<^sub>R x) ` S)"
 proof (cases "c = 0")
   case True then show ?thesis
     by (auto simp: image_constant_conv)
 next
   case False
   from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) -` S)"
     by (simp add: continuous_closed_vimage)
   also have "(\<lambda>x. inverse c *\<^sub>R x) -` S = (\<lambda>x. c *\<^sub>R x) ` S"
     using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
   finally show ?thesis .
 qed
 
 lemma closed_negations:
   fixes S :: "'a::real_normed_vector set"
   assumes "closed S"
   shows "closed ((\<lambda>x. -x) ` S)"
   using closed_scaling[OF assms, of "- 1"] by simp
 
 lemma compact_closed_sums:
   fixes S :: "'a::real_normed_vector set"
   assumes "compact S" and "closed T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"
   {
     fix x l
     assume as: "\<forall>n. x n \<in> ?S"  "(x \<longlongrightarrow> l) sequentially"
     from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)"  "\<forall>n. fst (f n) \<in> S"  "\<forall>n. snd (f n) \<in> T"
       using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto
     obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
       using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
     have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
       using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
       unfolding o_def
       by auto
     then have "l - l' \<in> T"
       using assms(2)[unfolded closed_sequential_limits,
         THEN spec[where x="\<lambda> n. snd (f (r n))"],
         THEN spec[where x="l - l'"]]
       using f(3)
       by auto
     then have "l \<in> ?S"
-      using \<open>l' \<in> S\<close>
-      apply auto
-      apply (rule_tac x=l' in exI)
-      apply (rule_tac x="l - l'" in exI, auto)
-      done
+      using \<open>l' \<in> S\<close> by force
   }
   moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by force
   ultimately show ?thesis
     unfolding closed_sequential_limits
     by (metis (no_types, lifting))
 qed
 
 lemma closed_compact_sums:
   fixes S T :: "'a::real_normed_vector set"
   assumes "closed S" "compact T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
 proof -
   have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
     by auto
   then show ?thesis
     using compact_closed_sums[OF assms(2,1)] by simp
 qed
 
 lemma compact_closed_differences:
   fixes S T :: "'a::real_normed_vector set"
   assumes "compact S" "closed T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
     by force
   then show ?thesis
-    using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
+    by (metis assms closed_negations compact_closed_sums)
 qed
 
 lemma closed_compact_differences:
   fixes S T :: "'a::real_normed_vector set"
   assumes "closed S" "compact T"
   shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
 proof -
   have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
     by auto
  then show ?thesis
   using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
 qed
 
 lemma closed_translation:
   "closed ((+) a ` S)" if "closed S" for a :: "'a::real_normed_vector"
 proof -
   have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = ((+) a ` S)" by auto
   then show ?thesis
     using compact_closed_sums [OF compact_sing [of a] that] by auto
 qed
 
 lemma closed_translation_subtract:
   "closed ((\<lambda>x. x - a) ` S)" if "closed S" for a :: "'a::real_normed_vector"
   using that closed_translation [of S "- a"] by (simp cong: image_cong_simp)
 
 lemma closure_translation:
   "closure ((+) a ` s) = (+) a ` closure s" for a :: "'a::real_normed_vector"
 proof -
   have *: "(+) a ` (- s) = - (+) a ` s"
     by (auto intro!: image_eqI [where x = "x - a" for x])
   show ?thesis
     using interior_translation [of a "- s", symmetric]
     by (simp add: closure_interior translation_Compl *)
 qed
 
 lemma closure_translation_subtract:
   "closure ((\<lambda>x. x - a) ` s) = (\<lambda>x. x - a) ` closure s" for a :: "'a::real_normed_vector"
   using closure_translation [of "- a" s] by (simp cong: image_cong_simp)
 
 lemma frontier_translation:
   "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
   by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
 
 lemma frontier_translation_subtract:
   "frontier ((+) a ` s) = (+) a ` frontier s" for a :: "'a::real_normed_vector"
   by (auto simp add: frontier_def translation_diff interior_translation closure_translation)
 
 lemma sphere_translation:
   "sphere (a + c) r = (+) a ` sphere c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma sphere_translation_subtract:
   "sphere (c - a) r = (\<lambda>x. x - a) ` sphere c r" for a :: "'n::real_normed_vector"
   using sphere_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 lemma cball_translation:
   "cball (a + c) r = (+) a ` cball c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma cball_translation_subtract:
   "cball (c - a) r = (\<lambda>x. x - a) ` cball c r" for a :: "'n::real_normed_vector"
   using cball_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 lemma ball_translation:
   "ball (a + c) r = (+) a ` ball c r" for a :: "'n::real_normed_vector"
   by (auto simp: dist_norm algebra_simps intro!: image_eqI [where x = "x - a" for x])
 
 lemma ball_translation_subtract:
   "ball (c - a) r = (\<lambda>x. x - a) ` ball c r" for a :: "'n::real_normed_vector"
   using ball_translation [of "- a" c] by (simp cong: image_cong_simp)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Homeomorphisms\<close>
 
 lemma homeomorphic_scaling:
-  fixes s :: "'a::real_normed_vector set"
+  fixes S :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
-  shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
+  shows "S homeomorphic ((\<lambda>x. c *\<^sub>R x) ` S)"
   unfolding homeomorphic_minimal
   apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
   apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
-  using assms
-  apply (auto simp: continuous_intros)
-  done
+  using assms by (auto simp: continuous_intros)
 
 lemma homeomorphic_translation:
-  fixes s :: "'a::real_normed_vector set"
-  shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
+  fixes S :: "'a::real_normed_vector set"
+  shows "S homeomorphic ((\<lambda>x. a + x) ` S)"
   unfolding homeomorphic_minimal
   apply (rule_tac x="\<lambda>x. a + x" in exI)
   apply (rule_tac x="\<lambda>x. -a + x" in exI)
-  using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
-    continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
-  apply auto
-  done
+  by (auto simp: continuous_intros)
 
 lemma homeomorphic_affinity:
-  fixes s :: "'a::real_normed_vector set"
+  fixes S :: "'a::real_normed_vector set"
   assumes "c \<noteq> 0"
-  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
+  shows "S homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` S)"
 proof -
-  have *: "(+) a ` (*\<^sub>R) c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
+  have *: "(+) a ` (*\<^sub>R) c ` S = (\<lambda>x. a + c *\<^sub>R x) ` S" by auto
   show ?thesis
-    using homeomorphic_trans
-    using homeomorphic_scaling[OF assms, of s]
-    using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a]
-    unfolding *
-    by auto
+    by (metis "*" assms homeomorphic_scaling homeomorphic_trans homeomorphic_translation)
 qed
 
 lemma homeomorphic_balls:
   fixes a b ::"'a::real_normed_vector"
   assumes "0 < d"  "0 < e"
   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
     and "(cball a d) homeomorphic (cball b e)" (is ?cth)
 proof -
   show ?th unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
-    done
+    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
   show ?cth unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
-    done
+    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_le_eq)
 qed
 
 lemma homeomorphic_spheres:
   fixes a b ::"'a::real_normed_vector"
   assumes "0 < d"  "0 < e"
   shows "(sphere a d) homeomorphic (sphere b e)"
 unfolding homeomorphic_minimal
     apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
     apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
     using assms
-    apply (auto intro!: continuous_intros
-      simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
-    done
+    by (auto intro!: continuous_intros simp: dist_commute dist_norm pos_divide_less_eq)
 
 lemma homeomorphic_ball01_UNIV:
   "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
   (is "?B homeomorphic ?U")
 proof
   have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
     apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
      apply (auto simp: field_split_simps)
     using norm_ge_zero [of x] apply linarith+
     done
   then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
     by blast
   have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
-    apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)
-    using that apply (auto simp: field_split_simps)
-    done
+    using that
+    by (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI) (auto simp: field_split_simps)
   then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
     by (force simp: field_split_simps dest: add_less_zeroD)
   show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
     by (rule continuous_intros | force)+
-  show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
-    apply (intro continuous_intros)
-    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
-    done
+  have 0: "\<And>z. 1 + norm z \<noteq> 0"
+    by (metis (no_types) le_add_same_cancel1 norm_ge_zero not_one_le_zero)
+  then show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
+    by (auto intro!: continuous_intros)
   show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
          x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
     by (auto simp: field_split_simps)
   show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
-    apply (auto simp: field_split_simps)
-    apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
-    done
+    using 0 by (auto simp: field_split_simps)
 qed
 
 proposition homeomorphic_ball_UNIV:
   fixes a ::"'a::real_normed_vector"
   assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
   using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Discrete\<close>
 
 lemma finite_implies_discrete:
   fixes S :: "'a::topological_space set"
   assumes "finite (f ` S)"
   shows "(\<forall>x \<in> S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x))"
 proof -
   have "\<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" if "x \<in> S" for x
   proof (cases "f ` S - {f x} = {}")
     case True
     with zero_less_numeral show ?thesis
       by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
   next
     case False
-    then obtain z where z: "z \<in> S" "f z \<noteq> f x"
+    then obtain z where "z \<in> S" "f z \<noteq> f x"
       by blast
-    have finn: "finite {norm (z - f x) |z. z \<in> f ` S - {f x}}"
+    moreover have finn: "finite {norm (z - f x) |z. z \<in> f ` S - {f x}}"
       using assms by simp
-    then have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` S - {f x}}"
-      apply (rule finite_imp_less_Inf)
-      using z apply force+
-      done
+    ultimately have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` S - {f x}}"
+      by (force intro: finite_imp_less_Inf)
     show ?thesis
       by (force intro!: * cInf_le_finite [OF finn])
   qed
   with assms show ?thesis
     by blast
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Completeness of "Isometry" (up to constant bounds)\<close>
 
 lemma cauchy_isometric:\<comment> \<open>TODO: rename lemma to \<open>Cauchy_isometric\<close>\<close>
   assumes e: "e > 0"
     and s: "subspace s"
     and f: "bounded_linear f"
     and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
     and xs: "\<forall>n. x n \<in> s"
     and cf: "Cauchy (f \<circ> x)"
   shows "Cauchy x"
 proof -
   interpret f: bounded_linear f by fact
   have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real
   proof -
     from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
       using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
       by auto
     have "norm (x n - x N) < d" if "n \<ge> N" for n
     proof -
       have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
         using subspace_diff[OF s, of "x n" "x N"]
         using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
         using normf[THEN bspec[where x="x n - x N"]]
         by auto
       also have "norm (f (x n - x N)) < e * d"
         using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
       finally show ?thesis
         using \<open>e>0\<close> by simp
     qed
     then show ?thesis by auto
   qed
   then show ?thesis
     by (simp add: Cauchy_altdef2 dist_norm)
 qed
 
 lemma complete_isometric_image:
   assumes "0 < e"
     and s: "subspace s"
     and f: "bounded_linear f"
     and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
     and cs: "complete s"
   shows "complete (f ` s)"
 proof -
   have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
     if as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g" for g
   proof -
     from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
       using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
     then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
     then have "f \<circ> x = g" by (simp add: fun_eq_iff)
     then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
       using cs[unfolded complete_def, THEN spec[where x=x]]
       using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
       by auto
     then show ?thesis
       using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
       by (auto simp: \<open>f \<circ> x = g\<close>)
   qed
   then show ?thesis
     unfolding complete_def by auto
 qed
 
 subsection \<open>Connected Normed Spaces\<close>
 
 lemma compact_components:
   fixes s :: "'a::heine_borel set"
   shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"
 by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)
 
 lemma discrete_subset_disconnected:
   fixes S :: "'a::topological_space set"
   fixes t :: "'b::real_normed_vector set"
   assumes conf: "continuous_on S f"
       and no: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
    shows "f ` S \<subseteq> {y. connected_component_set (f ` S) y = {y}}"
 proof -
   { fix x assume x: "x \<in> S"
     then obtain e where "e>0" and ele: "\<And>y. \<lbrakk>y \<in> S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
       using conf no [OF x] by auto
-    then have e2: "0 \<le> e / 2"
+    then have e2: "0 \<le> e/2"
       by simp
-    have "f y = f x" if "y \<in> S" and ccs: "f y \<in> connected_component_set (f ` S) (f x)" for y
-      apply (rule ccontr)
-      using connected_closed [of "connected_component_set (f ` S) (f x)"] \<open>e>0\<close>
-      apply (simp add: del: ex_simps)
-      apply (drule spec [where x="cball (f x) (e / 2)"])
-      apply (drule spec [where x="- ball(f x) e"])
-      apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)
-        apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
-       using centre_in_cball connected_component_refl_eq e2 x apply blast
-      using ccs
-      apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF \<open>y \<in> S\<close>])
-      done
+    define F where "F \<equiv> connected_component_set (f ` S) (f x)"
+    have False if "y \<in> S" and ccs: "f y \<in> F" and not: "f y \<noteq> f x" for y 
+    proof -
+      define C where "C \<equiv> cball (f x) (e/2)"
+      define D where "D \<equiv> - ball (f x) e"
+      have disj: "C \<inter> D = {}"
+        unfolding C_def D_def using \<open>0 < e\<close> by fastforce
+      moreover have FCD: "F \<subseteq> C \<union> D"
+      proof -
+        have "t \<in> C \<or> t \<in> D" if "t \<in> F" for t
+        proof -
+          obtain y where "y \<in> S" "t = f y"
+            using F_def \<open>t \<in> F\<close> connected_component_in by blast
+          then show ?thesis
+            by (metis C_def ComplI D_def centre_in_cball dist_norm e2 ele mem_ball norm_minus_commute not_le)
+        qed
+        then show ?thesis
+          by auto
+      qed
+      ultimately have "C \<inter> F = {} \<or> D \<inter> F = {}"
+        using connected_closed [of "F"] \<open>e>0\<close> not
+        unfolding C_def D_def
+        by (metis Elementary_Metric_Spaces.open_ball F_def closed_cball connected_connected_component inf_bot_left open_closed)
+      moreover have "C \<inter> F \<noteq> {}"
+        unfolding disjoint_iff
+        by (metis FCD ComplD image_eqI mem_Collect_eq subsetD x  D_def F_def Un_iff \<open>0 < e\<close> centre_in_ball connected_component_refl_eq)
+      moreover have "D \<inter> F \<noteq> {}"
+        unfolding disjoint_iff
+        by (metis ComplI D_def ccs dist_norm ele mem_ball norm_minus_commute not not_le that(1))
+      ultimately show ?thesis by metis
+    qed
     moreover have "connected_component_set (f ` S) (f x) \<subseteq> f ` S"
       by (auto simp: connected_component_in)
     ultimately have "connected_component_set (f ` S) (f x) = {f x}"
-      by (auto simp: x)
+      by (auto simp: x F_def)
   }
   with assms show ?thesis
     by blast
 qed
 
 lemma continuous_disconnected_range_constant_eq:
       "(connected S \<longleftrightarrow>
            (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
             \<forall>t. continuous_on S f \<and> f ` S \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})
             \<longrightarrow> f constant_on S))" (is ?thesis1)
   and continuous_discrete_range_constant_eq:
       "(connected S \<longleftrightarrow>
          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
           continuous_on S f \<and>
           (\<forall>x \<in> S. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> S \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))
           \<longrightarrow> f constant_on S))" (is ?thesis2)
   and continuous_finite_range_constant_eq:
       "(connected S \<longleftrightarrow>
          (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
           continuous_on S f \<and> finite (f ` S)
           \<longrightarrow> f constant_on S))" (is ?thesis3)
 proof -
   have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>
     \<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"
     by blast
   have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"
     apply (rule *)
     using continuous_disconnected_range_constant apply metis
     apply clarify
     apply (frule discrete_subset_disconnected; blast)
     apply (blast dest: finite_implies_discrete)
     apply (blast intro!: finite_range_constant_imp_connected)
     done
   then show ?thesis1 ?thesis2 ?thesis3
     by blast+
 qed
 
 lemma continuous_discrete_range_constant:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
   assumes S: "connected S"
       and "continuous_on S f"
       and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
     shows "f constant_on S"
   using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms by blast
 
 lemma continuous_finite_range_constant:
   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
   assumes "connected S"
       and "continuous_on S f"
       and "finite (f ` S)"
     shows "f constant_on S"
   using assms continuous_finite_range_constant_eq  by blast
 
 end