
diff -r ea4dc7603f0b -r 371f023d3dbd src/HOL/Real/HahnBanach/HahnBanach.thy
--- a/src/HOL/Real/HahnBanach/HahnBanach.thy	Sun Jun 04 00:09:04 2000 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanach.thy	Sun Jun 04 19:39:29 2000 +0200
@@ -3,17 +3,17 @@
     Author:     Gertrud Bauer, TU Munich
 *)
 
-header {* The Hahn-Banach Theorem *};
+header {* The Hahn-Banach Theorem *}
 
 theory HahnBanach
-     = HahnBanachSupLemmas + HahnBanachExtLemmas + ZornLemma:;
+     = HahnBanachSupLemmas + HahnBanachExtLemmas + ZornLemma:
 
 text {*
   We present the proof of two different versions of the Hahn-Banach 
   Theorem, closely following \cite[\S36]{Heuser:1986}.
-*};
+*}
 
-subsection {* The Hahn-Banach Theorem for vector spaces *};
+subsection {* The Hahn-Banach Theorem for vector spaces *}
 
 text {* {\bf Theorem.} Let $f$ be a linear form defined on a subspace 
  $F$ of a real vector space $E$, such that $f$ is bounded by a seminorm 
@@ -32,282 +32,282 @@
  a norm-preserving way to a greater vector space $H_0$. 
  So $g$ cannot be maximal in $M$.
  \bigskip
-*};
+*}
 
 theorem HahnBanach: "[| is_vectorspace E; is_subspace F E; 
  is_seminorm E p; is_linearform F f; ALL x:F. f x <= p x |]
   ==> EX h. is_linearform E h & (ALL x:F. h x = f x)
-          & (ALL x:E. h x <= p x)";
-proof -;
+          & (ALL x:E. h x <= p x)"
+proof -
 
-txt {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$ and $f$ a linear form on $F$ such that $f$ is bounded by $p$. *};
+txt {* Let $E$ be a vector space, $F$ a subspace of $E$, $p$ a seminorm on $E$ and $f$ a linear form on $F$ such that $f$ is bounded by $p$. *}
 
   assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p" 
-    "is_linearform F f" "ALL x:F. f x <= p x";
+    "is_linearform F f" "ALL x:F. f x <= p x"
 
-txt {* Define $M$ as the set of all norm-preserving extensions of $F$.  *};
+txt {* Define $M$ as the set of all norm-preserving extensions of $F$.  *}
 
-  def M == "norm_pres_extensions E p F f";
-  {;
-    fix c; assume "c : chain M" "EX x. x:c";
+  def M == "norm_pres_extensions E p F f"
+  {
+    fix c assume "c : chain M" "EX x. x:c"
 
-txt {* Show that every non-empty chain $c$ in $M$ has an upper bound in $M$: $\Union c$ is greater that every element of the chain $c$, so $\Union c$ is an upper bound of $c$ that lies in $M$. *};
+txt {* Show that every non-empty chain $c$ in $M$ has an upper bound in $M$: $\Union c$ is greater that every element of the chain $c$, so $\Union c$ is an upper bound of $c$ that lies in $M$. *}
 
-    have "Union c : M";
-    proof (unfold M_def, rule norm_pres_extensionI);
+    have "Union c : M"
+    proof (unfold M_def, rule norm_pres_extensionI)
       show "EX (H::'a set) h::'a => real. graph H h = Union c
               & is_linearform H h 
               & is_subspace H E 
               & is_subspace F H 
               & graph F f <= graph H h
-              & (ALL x::'a:H. h x <= p x)";
-      proof (intro exI conjI);
-        let ?H = "domain (Union c)";
-        let ?h = "funct (Union c)";
+              & (ALL x::'a:H. h x <= p x)"
+      proof (intro exI conjI)
+        let ?H = "domain (Union c)"
+        let ?h = "funct (Union c)"
 
-        show a: "graph ?H ?h = Union c"; 
-        proof (rule graph_domain_funct);
-          fix x y z; assume "(x, y) : Union c" "(x, z) : Union c";
-          show "z = y"; by (rule sup_definite);
-        qed;
-        show "is_linearform ?H ?h"; 
-          by (simp! add: sup_lf a);
-        show "is_subspace ?H E";
-          by (rule sup_subE [OF _ _ _ a]) (simp!)+;
-        show "is_subspace F ?H"; 
-          by (rule sup_supF [OF _ _ _ a]) (simp!)+;
-        show "graph F f <= graph ?H ?h"; 
-          by (rule sup_ext [OF _ _ _ a]) (simp!)+;
-        show "ALL x::'a:?H. ?h x <= p x"; 
-          by (rule sup_norm_pres [OF _ _ a]) (simp!)+;
-      qed;
-    qed;
-  };
+        show a: "graph ?H ?h = Union c" 
+        proof (rule graph_domain_funct)
+          fix x y z assume "(x, y) : Union c" "(x, z) : Union c"
+          show "z = y" by (rule sup_definite)
+        qed
+        show "is_linearform ?H ?h" 
+          by (simp! add: sup_lf a)
+        show "is_subspace ?H E"
+          by (rule sup_subE [OF _ _ _ a]) (simp!)+
+        show "is_subspace F ?H" 
+          by (rule sup_supF [OF _ _ _ a]) (simp!)+
+        show "graph F f <= graph ?H ?h" 
+          by (rule sup_ext [OF _ _ _ a]) (simp!)+
+        show "ALL x::'a:?H. ?h x <= p x" 
+          by (rule sup_norm_pres [OF _ _ a]) (simp!)+
+      qed
+    qed
+  }
   
-txt {* With Zorn's Lemma we can conclude that there is a maximal element $g$ in $M$. *};
+txt {* With Zorn's Lemma we can conclude that there is a maximal element $g$ in $M$. *}
 
-  hence "EX g:M. ALL x:M. g <= x --> g = x";
-  proof (rule Zorn's_Lemma);
-    txt {* We show that $M$ is non-empty: *};
-    have "graph F f : norm_pres_extensions E p F f";
-    proof (rule norm_pres_extensionI2);
-      have "is_vectorspace F"; ..;
-      thus "is_subspace F F"; ..;
-    qed (blast!)+; 
-    thus "graph F f : M"; by (simp!);
-  qed;
-  thus ?thesis;
-  proof;
+  hence "EX g:M. ALL x:M. g <= x --> g = x"
+  proof (rule Zorn's_Lemma)
+    txt {* We show that $M$ is non-empty: *}
+    have "graph F f : norm_pres_extensions E p F f"
+    proof (rule norm_pres_extensionI2)
+      have "is_vectorspace F" ..
+      thus "is_subspace F F" ..
+    qed (blast!)+ 
+    thus "graph F f : M" by (simp!)
+  qed
+  thus ?thesis
+  proof
 
-txt {* We take this maximal element $g$.  *};
+txt {* We take this maximal element $g$.  *}
 
-    fix g; assume "g:M" "ALL x:M. g <= x --> g = x";
-    show ?thesis;
+    fix g assume "g:M" "ALL x:M. g <= x --> g = x"
+    show ?thesis
 
       txt {* $g$ is a norm-preserving extension of $f$, that is: $g$
       is the graph of a linear form $h$, defined on a subspace $H$ of
       $E$, which is a superspace of $F$. $h$ is an extension of $f$
-      and $h$ is again bounded by $p$. *};
+      and $h$ is again bounded by $p$. *}
 
       obtain H h where "graph H h = g" "is_linearform H h" 
         "is_subspace H E" "is_subspace F H" "graph F f <= graph H h" 
-        "ALL x:H. h x <= p x";
-      proof -;
+        "ALL x:H. h x <= p x"
+      proof -
         have "EX H h. graph H h = g & is_linearform H h 
           & is_subspace H E & is_subspace F H
           & graph F f <= graph H h
-          & (ALL x:H. h x <= p x)"; by (simp! add: norm_pres_extension_D);
-        thus ?thesis; by (elim exE conjE) rule;
-      qed;
-      have h: "is_vectorspace H"; ..;
+          & (ALL x:H. h x <= p x)" by (simp! add: norm_pres_extension_D)
+        thus ?thesis by (elim exE conjE) rule
+      qed
+      have h: "is_vectorspace H" ..
 
-txt {* We show that $h$ is defined on whole $E$ by classical contradiction.  *}; 
+txt {* We show that $h$ is defined on whole $E$ by classical contradiction.  *} 
 
-      have "H = E"; 
-      proof (rule classical);
+      have "H = E" 
+      proof (rule classical)
 
-	txt {* Assume $h$ is not defined on whole $E$. *};
+	txt {* Assume $h$ is not defined on whole $E$. *}
 
-        assume "H ~= E";
+        assume "H ~= E"
 
-txt {* Then show that $h$ can be extended in a norm-preserving way to a function $h_0$ with the graph $g_{h0}$.  *};
+txt {* Then show that $h$ can be extended in a norm-preserving way to a function $h_0$ with the graph $g_{h0}$.  *}
 
-        have "EX g_h0 : M. g <= g_h0 & g ~= g_h0"; 
+        have "EX g_h0 : M. g <= g_h0 & g ~= g_h0" 
 
-	  txt {* Consider $x_0 \in E \setminus H$. *};
+	  txt {* Consider $x_0 \in E \setminus H$. *}
 
-          obtain x0 where "x0:E" "x0~:H"; 
-          proof -;
-            have "EX x0:E. x0~:H";
-            proof (rule set_less_imp_diff_not_empty);
-              have "H <= E"; ..;
-              thus "H < E"; ..;
-            qed;
-            thus ?thesis; by blast;
-          qed;
-          have x0: "x0 ~= 00";
-          proof (rule classical);
-            presume "x0 = 00";
-            with h; have "x0:H"; by simp;
-            thus ?thesis; by contradiction;
-          qed blast;
+          obtain x0 where "x0:E" "x0~:H" 
+          proof -
+            have "EX x0:E. x0~:H"
+            proof (rule set_less_imp_diff_not_empty)
+              have "H <= E" ..
+              thus "H < E" ..
+            qed
+            thus ?thesis by blast
+          qed
+          have x0: "x0 ~= 00"
+          proof (rule classical)
+            presume "x0 = 00"
+            with h have "x0:H" by simp
+            thus ?thesis by contradiction
+          qed blast
 
-txt {* Define $H_0$ as the direct sum of $H$ and the linear closure of $x_0$.  *};
+txt {* Define $H_0$ as the direct sum of $H$ and the linear closure of $x_0$.  *}
 
-          def H0 == "H + lin x0";
-          show ?thesis;
+          def H0 == "H + lin x0"
+          show ?thesis
 
 	    txt {* Pick a real number $\xi$ that fulfills certain
 	    inequations, which will be used to establish that $h_0$ is
-	    a norm-preserving extension of $h$. *};
+	    a norm-preserving extension of $h$. *}
 
             obtain xi where "ALL y:H. - p (y + x0) - h y <= xi 
-                              & xi <= p (y + x0) - h y";
-            proof -;
-	      from h; have "EX xi. ALL y:H. - p (y + x0) - h y <= xi 
-                              & xi <= p (y + x0) - h y"; 
-              proof (rule ex_xi);
-                fix u v; assume "u:H" "v:H";
-                from h; have "h v - h u = h (v - u)";
-                  by (simp! add: linearform_diff);
-                also; have "... <= p (v - u)";
-                  by (simp!);
-                also; have "v - u = x0 + - x0 + v + - u";
-                  by (simp! add: diff_eq1);
-                also; have "... = v + x0 + - (u + x0)";
-                  by (simp!);
-                also; have "... = (v + x0) - (u + x0)";
-                  by (simp! add: diff_eq1);
-                also; have "p ... <= p (v + x0) + p (u + x0)";
-                  by (rule seminorm_diff_subadditive) (simp!)+;
-                finally; have "h v - h u <= p (v + x0) + p (u + x0)"; .;
+                              & xi <= p (y + x0) - h y"
+            proof -
+	      from h have "EX xi. ALL y:H. - p (y + x0) - h y <= xi 
+                              & xi <= p (y + x0) - h y" 
+              proof (rule ex_xi)
+                fix u v assume "u:H" "v:H"
+                from h have "h v - h u = h (v - u)"
+                  by (simp! add: linearform_diff)
+                also have "... <= p (v - u)"
+                  by (simp!)
+                also have "v - u = x0 + - x0 + v + - u"
+                  by (simp! add: diff_eq1)
+                also have "... = v + x0 + - (u + x0)"
+                  by (simp!)
+                also have "... = (v + x0) - (u + x0)"
+                  by (simp! add: diff_eq1)
+                also have "p ... <= p (v + x0) + p (u + x0)"
+                  by (rule seminorm_diff_subadditive) (simp!)+
+                finally have "h v - h u <= p (v + x0) + p (u + x0)" .
 
-                thus "- p (u + x0) - h u <= p (v + x0) - h v";
-                  by (rule real_diff_ineq_swap);
-              qed;
-              thus ?thesis; by rule rule;
-            qed;
+                thus "- p (u + x0) - h u <= p (v + x0) - h v"
+                  by (rule real_diff_ineq_swap)
+              qed
+              thus ?thesis by rule rule
+            qed
 
-txt {* Define the extension $h_0$ of $h$ to $H_0$ using $\xi$.  *};
+txt {* Define the extension $h_0$ of $h$ to $H_0$ using $\xi$.  *}
 
             def h0 == "\\<lambda>x. let (y,a) = SOME (y, a). x = y + a (*) x0 
                                                   & y:H
-                           in (h y) + a * xi";
-            show ?thesis;
-            proof;
+                           in (h y) + a * xi"
+            show ?thesis
+            proof
  
-txt {* Show that $h_0$ is an extension of $h$  *};
+txt {* Show that $h_0$ is an extension of $h$  *}
  
-              show "g <= graph H0 h0 & g ~= graph H0 h0";
-              proof;
-		show "g <= graph H0 h0";
-		proof -;
-		  have  "graph H h <= graph H0 h0";
-                  proof (rule graph_extI);
-		    fix t; assume "t:H"; 
+              show "g <= graph H0 h0 & g ~= graph H0 h0"
+              proof
+		show "g <= graph H0 h0"
+		proof -
+		  have  "graph H h <= graph H0 h0"
+                  proof (rule graph_extI)
+		    fix t assume "t:H" 
 		    have "(SOME (y, a). t = y + a (*) x0 & y : H)
-                         = (t,#0)";
-		      by (rule decomp_H0_H [OF _ _ _ _ _ x0]);
-		    thus "h t = h0 t"; by (simp! add: Let_def);
-		  next;
-		    show "H <= H0";
-		    proof (rule subspace_subset);
-		      show "is_subspace H H0";
-		      proof (unfold H0_def, rule subspace_vs_sum1);
-			show "is_vectorspace H"; ..;
-			show "is_vectorspace (lin x0)"; ..;
-		      qed;
-		    qed;
-		  qed; 
-		  thus ?thesis; by (simp!);
-		qed;
-                show "g ~= graph H0 h0";
-		proof -;
-		  have "graph H h ~= graph H0 h0";
-		  proof;
-		    assume e: "graph H h = graph H0 h0";
-		    have "x0 : H0"; 
-		    proof (unfold H0_def, rule vs_sumI);
-		      show "x0 = 00 + x0"; by (simp!);
-		      from h; show "00 : H"; ..;
-		      show "x0 : lin x0"; by (rule x_lin_x);
-		    qed;
-		    hence "(x0, h0 x0) : graph H0 h0"; ..;
-		    with e; have "(x0, h0 x0) : graph H h"; by simp;
-		    hence "x0 : H"; ..;
-		    thus False; by contradiction;
-		  qed;
-		  thus ?thesis; by (simp!);
-		qed;
-              qed;
+                         = (t,#0)"
+		      by (rule decomp_H0_H [OF _ _ _ _ _ x0])
+		    thus "h t = h0 t" by (simp! add: Let_def)
+		  next
+		    show "H <= H0"
+		    proof (rule subspace_subset)
+		      show "is_subspace H H0"
+		      proof (unfold H0_def, rule subspace_vs_sum1)
+			show "is_vectorspace H" ..
+			show "is_vectorspace (lin x0)" ..
+		      qed
+		    qed
+		  qed 
+		  thus ?thesis by (simp!)
+		qed
+                show "g ~= graph H0 h0"
+		proof -
+		  have "graph H h ~= graph H0 h0"
+		  proof
+		    assume e: "graph H h = graph H0 h0"
+		    have "x0 : H0" 
+		    proof (unfold H0_def, rule vs_sumI)
+		      show "x0 = 00 + x0" by (simp!)
+		      from h show "00 : H" ..
+		      show "x0 : lin x0" by (rule x_lin_x)
+		    qed
+		    hence "(x0, h0 x0) : graph H0 h0" ..
+		    with e have "(x0, h0 x0) : graph H h" by simp
+		    hence "x0 : H" ..
+		    thus False by contradiction
+		  qed
+		  thus ?thesis by (simp!)
+		qed
+              qed
 	      
-txt {* and $h_0$ is norm-preserving.  *}; 
+txt {* and $h_0$ is norm-preserving.  *} 
 
-              show "graph H0 h0 : M";
-	      proof -;
-		have "graph H0 h0 : norm_pres_extensions E p F f";
-		proof (rule norm_pres_extensionI2);
-		  show "is_linearform H0 h0";
-		    by (rule h0_lf [OF _ _ _ _ _ _ x0]) (simp!)+;
-		  show "is_subspace H0 E";
-		    by (unfold H0_def) (rule vs_sum_subspace [OF _ lin_subspace]);
-		  have "is_subspace F H"; .;
-		  also; from h lin_vs; 
-		  have [fold H0_def]: "is_subspace H (H + lin x0)"; ..;
-		  finally (subspace_trans [OF _ h]); 
-		  show f_h0: "is_subspace F H0"; .;
+              show "graph H0 h0 : M"
+	      proof -
+		have "graph H0 h0 : norm_pres_extensions E p F f"
+		proof (rule norm_pres_extensionI2)
+		  show "is_linearform H0 h0"
+		    by (rule h0_lf [OF _ _ _ _ _ _ x0]) (simp!)+
+		  show "is_subspace H0 E"
+		    by (unfold H0_def) (rule vs_sum_subspace [OF _ lin_subspace])
+		  have "is_subspace F H" .
+		  also from h lin_vs 
+		  have [fold H0_def]: "is_subspace H (H + lin x0)" ..
+		  finally (subspace_trans [OF _ h]) 
+		  show f_h0: "is_subspace F H0" .
 		
-		  show "graph F f <= graph H0 h0";
-		  proof (rule graph_extI);
-		    fix x; assume "x:F";
-		    have "f x = h x"; ..;
-		    also; have " ... = h x + #0 * xi"; by simp;
-		    also; have "... = (let (y,a) = (x, #0) in h y + a * xi)";
-		      by (simp add: Let_def);
-		    also; have 
-		      "(x, #0) = (SOME (y, a). x = y + a (*) x0 & y : H)";
-		      by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x0]) (simp!)+;
-		    also; have 
+		  show "graph F f <= graph H0 h0"
+		  proof (rule graph_extI)
+		    fix x assume "x:F"
+		    have "f x = h x" ..
+		    also have " ... = h x + #0 * xi" by simp
+		    also have "... = (let (y,a) = (x, #0) in h y + a * xi)"
+		      by (simp add: Let_def)
+		    also have 
+		      "(x, #0) = (SOME (y, a). x = y + a (*) x0 & y : H)"
+		      by (rule decomp_H0_H [RS sym, OF _ _ _ _ _ x0]) (simp!)+
+		    also have 
 		      "(let (y,a) = (SOME (y,a). x = y + a (*) x0 & y : H)
                         in h y + a * xi) 
-                      = h0 x"; by (simp!);
-		    finally; show "f x = h0 x"; .;
-		  next;
-		    from f_h0; show "F <= H0"; ..;
-		  qed;
+                      = h0 x" by (simp!)
+		    finally show "f x = h0 x" .
+		  next
+		    from f_h0 show "F <= H0" ..
+		  qed
 		
-		  show "ALL x:H0. h0 x <= p x";
-		    by (rule h0_norm_pres [OF _ _ _ _ x0]);
-		qed;
-		thus "graph H0 h0 : M"; by (simp!);
-	      qed;
-            qed;
-          qed;
-        qed;
+		  show "ALL x:H0. h0 x <= p x"
+		    by (rule h0_norm_pres [OF _ _ _ _ x0])
+		qed
+		thus "graph H0 h0 : M" by (simp!)
+	      qed
+            qed
+          qed
+        qed
 
-txt {* So the graph $g$ of $h$ cannot be maximal. Contradiction.  *}; 
+txt {* So the graph $g$ of $h$ cannot be maximal. Contradiction.  *} 
 
-        hence "~ (ALL x:M. g <= x --> g = x)"; by simp;
-        thus ?thesis; by contradiction;
-      qed; 
+        hence "~ (ALL x:M. g <= x --> g = x)" by simp
+        thus ?thesis by contradiction
+      qed 
 
 txt {* Now we have a linear extension $h$ of $f$ to $E$ that is 
-bounded by $p$. *};
+bounded by $p$. *}
 
       thus "EX h. is_linearform E h & (ALL x:F. h x = f x) 
-                & (ALL x:E. h x <= p x)";
-      proof (intro exI conjI);
-        assume eq: "H = E";
-	from eq; show "is_linearform E h"; by (simp!);
-	show "ALL x:F. h x = f x"; 
-	proof (intro ballI, rule sym);
-	  fix x; assume "x:F"; show "f x = h x "; ..;
-	qed;
-	from eq; show "ALL x:E. h x <= p x"; by (force!);
-      qed;
-    qed;
-  qed;
-qed;
+                & (ALL x:E. h x <= p x)"
+      proof (intro exI conjI)
+        assume eq: "H = E"
+	from eq show "is_linearform E h" by (simp!)
+	show "ALL x:F. h x = f x" 
+	proof (intro ballI, rule sym)
+	  fix x assume "x:F" show "f x = h x " ..
+	qed
+	from eq show "ALL x:E. h x <= p x" by (force!)
+      qed
+    qed
+  qed
+qed
 (*
 theorem HahnBanach: 
   "[| is_vectorspace E; is_subspace F E; is_seminorm E p; 
@@ -320,69 +320,69 @@
          "is_linearform F f" "ALL x:F. f x <= p x";
   
   txt{* We define $M$ to be the set of all linear extensions 
-  of $f$ to superspaces of $F$, which are bounded by $p$. *};
+  of $f$ to superspaces of $F$, which are bounded by $p$. *}
 
-  def M == "norm_pres_extensions E p F f";
+  def M == "norm_pres_extensions E p F f"
   
-  txt{* We show that $M$ is non-empty: *};
+  txt{* We show that $M$ is non-empty: *}
  
-  have aM: "graph F f : norm_pres_extensions E p F f";
-  proof (rule norm_pres_extensionI2);
-    have "is_vectorspace F"; ..;
-    thus "is_subspace F F"; ..;
-  qed (blast!)+;
+  have aM: "graph F f : norm_pres_extensions E p F f"
+  proof (rule norm_pres_extensionI2)
+    have "is_vectorspace F" ..
+    thus "is_subspace F F" ..
+  qed (blast!)+
 
-  subsubsect {* Existence of a limit function *}; 
+  subsubsect {* Existence of a limit function *} 
 
   txt {* For every non-empty chain of norm-preserving extensions
   the union of all functions in the chain is again a norm-preserving
   extension. (The union is an upper bound for all elements. 
   It is even the least upper bound, because every upper bound of $M$
-  is also an upper bound for $\Union c$, as $\Union c\in M$) *};
+  is also an upper bound for $\Union c$, as $\Union c\in M$) *}
 
-  {;
-    fix c; assume "c:chain M" "EX x. x:c";
-    have "Union c : M";
+  {
+    fix c assume "c:chain M" "EX x. x:c"
+    have "Union c : M"
 
-    proof (unfold M_def, rule norm_pres_extensionI);
+    proof (unfold M_def, rule norm_pres_extensionI)
       show "EX (H::'a set) h::'a => real. graph H h = Union c
               & is_linearform H h 
               & is_subspace H E 
               & is_subspace F H 
               & graph F f <= graph H h
-              & (ALL x::'a:H. h x <= p x)";
-      proof (intro exI conjI);
-        let ?H = "domain (Union c)";
-        let ?h = "funct (Union c)";
+              & (ALL x::'a:H. h x <= p x)"
+      proof (intro exI conjI)
+        let ?H = "domain (Union c)"
+        let ?h = "funct (Union c)"
 
-        show a: "graph ?H ?h = Union c"; 
-        proof (rule graph_domain_funct);
-          fix x y z; assume "(x, y) : Union c" "(x, z) : Union c";
-          show "z = y"; by (rule sup_definite);
-        qed;
-        show "is_linearform ?H ?h";  
-          by (simp! add: sup_lf a);
-        show "is_subspace ?H E"; 
-          by (rule sup_subE, rule a) (simp!)+;
-        show "is_subspace F ?H"; 
-          by (rule sup_supF, rule a) (simp!)+;
-        show "graph F f <= graph ?H ?h"; 
-          by (rule sup_ext, rule a) (simp!)+;
-        show "ALL x::'a:?H. ?h x <= p x"; 
-          by (rule sup_norm_pres, rule a) (simp!)+;
-      qed;
-    qed;
-  };
+        show a: "graph ?H ?h = Union c" 
+        proof (rule graph_domain_funct)
+          fix x y z assume "(x, y) : Union c" "(x, z) : Union c"
+          show "z = y" by (rule sup_definite)
+        qed
+        show "is_linearform ?H ?h"  
+          by (simp! add: sup_lf a)
+        show "is_subspace ?H E" 
+          by (rule sup_subE, rule a) (simp!)+
+        show "is_subspace F ?H" 
+          by (rule sup_supF, rule a) (simp!)+
+        show "graph F f <= graph ?H ?h" 
+          by (rule sup_ext, rule a) (simp!)+
+        show "ALL x::'a:?H. ?h x <= p x" 
+          by (rule sup_norm_pres, rule a) (simp!)+
+      qed
+    qed
+  }
  
   txt {* According to Zorn's Lemma there is
-  a maximal norm-preserving extension $g\in M$. *};
+  a maximal norm-preserving extension $g\in M$. *}
   
-  with aM; have bex_g: "EX g:M. ALL x:M. g <= x --> g = x";
-    by (simp! add: Zorn's_Lemma);
+  with aM have bex_g: "EX g:M. ALL x:M. g <= x --> g = x"
+    by (simp! add: Zorn's_Lemma)
 
-  thus ?thesis;
-  proof;
-    fix g; assume g: "g:M" "ALL x:M. g <= x --> g = x";
+  thus ?thesis
+  proof
+    fix g assume g: "g:M" "ALL x:M. g <= x --> g = x"
  
     have ex_Hh: 
       "EX H h. graph H h = g 
@@ -390,145 +390,145 @@
            & is_subspace H E 
            & is_subspace F H
            & graph F f <= graph H h
-           & (ALL x:H. h x <= p x) "; 
-      by (simp! add: norm_pres_extension_D);
-    thus ?thesis;
-    proof (elim exE conjE, intro exI);
-      fix H h; 
+           & (ALL x:H. h x <= p x) " 
+      by (simp! add: norm_pres_extension_D)
+    thus ?thesis
+    proof (elim exE conjE, intro exI)
+      fix H h 
       assume "graph H h = g" "is_linearform (H::'a set) h" 
              "is_subspace H E" "is_subspace F H"
         and h_ext: "graph F f <= graph H h"
-        and h_bound: "ALL x:H. h x <= p x";
+        and h_bound: "ALL x:H. h x <= p x"
 
-      have h: "is_vectorspace H"; ..;
-      have f: "is_vectorspace F"; ..;
+      have h: "is_vectorspace H" ..
+      have f: "is_vectorspace F" ..
 
-subsubsect {* The domain of the limit function *};
+subsubsect {* The domain of the limit function *}
 
-have eq: "H = E"; 
-proof (rule classical);
+have eq: "H = E" 
+proof (rule classical)
 
-txt {* Assume that the domain of the supremum is not $E$, *};
+txt {* Assume that the domain of the supremum is not $E$, *}
 
-  assume "H ~= E";
-  have "H <= E"; ..;
-  hence "H < E"; ..;
+  assume "H ~= E"
+  have "H <= E" ..
+  hence "H < E" ..
  
-  txt{* then there exists an element $x_0$ in $E \setminus H$: *};
+  txt{* then there exists an element $x_0$ in $E \setminus H$: *}
 
-  hence "EX x0:E. x0~:H"; 
-    by (rule set_less_imp_diff_not_empty);
+  hence "EX x0:E. x0~:H" 
+    by (rule set_less_imp_diff_not_empty)
 
   txt {* We get that $h$ can be extended  in a 
-  norm-preserving way to some $H_0$. *};
+  norm-preserving way to some $H_0$. *}
 
   hence "EX H0 h0. g <= graph H0 h0 & g ~= graph H0 h0 
-                 & graph H0 h0 : M";
-  proof;
-    fix x0; assume "x0:E" "x0~:H";
+                 & graph H0 h0 : M"
+  proof
+    fix x0 assume "x0:E" "x0~:H"
 
-    have x0: "x0 ~= 00";
-    proof (rule classical);
-      presume "x0 = 00";
-      with h; have "x0:H"; by simp;
-      thus ?thesis; by contradiction;
-    qed blast;
+    have x0: "x0 ~= 00"
+    proof (rule classical)
+      presume "x0 = 00"
+      with h have "x0:H" by simp
+      thus ?thesis by contradiction
+    qed blast
 
     txt {* Define $H_0$ as the (direct) sum of H and the 
-    linear closure of $x_0$. \label{ex-xi-use}*};
+    linear closure of $x_0$. \label{ex-xi-use}*}
 
-    def H0 == "H + lin x0";
+    def H0 == "H + lin x0"
 
-    from h; have xi: "EX xi. ALL y:H. - p (y + x0) - h y <= xi 
-                                    & xi <= p (y + x0) - h y";
-    proof (rule ex_xi);
-      fix u v; assume "u:H" "v:H";
-      from h; have "h v - h u = h (v - u)";
-         by (simp! add: linearform_diff);
-      also; from h_bound; have "... <= p (v - u)";
-        by (simp!);
-      also; have "v - u = x0 + - x0 + v + - u";
-        by (simp! add: diff_eq1);
-      also; have "... = v + x0 + - (u + x0)";
-        by (simp!);
-      also; have "... = (v + x0) - (u + x0)";
-        by (simp! add: diff_eq1);
-      also; have "p ... <= p (v + x0) + p (u + x0)";
-         by (rule seminorm_diff_subadditive) (simp!)+;
-      finally; have "h v - h u <= p (v + x0) + p (u + x0)"; .;
+    from h have xi: "EX xi. ALL y:H. - p (y + x0) - h y <= xi 
+                                    & xi <= p (y + x0) - h y"
+    proof (rule ex_xi)
+      fix u v assume "u:H" "v:H"
+      from h have "h v - h u = h (v - u)"
+         by (simp! add: linearform_diff)
+      also from h_bound have "... <= p (v - u)"
+        by (simp!)
+      also have "v - u = x0 + - x0 + v + - u"
+        by (simp! add: diff_eq1)
+      also have "... = v + x0 + - (u + x0)"
+        by (simp!)
+      also have "... = (v + x0) - (u + x0)"
+        by (simp! add: diff_eq1)
+      also have "p ... <= p (v + x0) + p (u + x0)"
+         by (rule seminorm_diff_subadditive) (simp!)+
+      finally have "h v - h u <= p (v + x0) + p (u + x0)" .
 
-      thus "- p (u + x0) - h u <= p (v + x0) - h v";
-        by (rule real_diff_ineq_swap);
-    qed;
+      thus "- p (u + x0) - h u <= p (v + x0) - h v"
+        by (rule real_diff_ineq_swap)
+    qed
     hence "EX h0. g <= graph H0 h0 & g ~= graph H0 h0
-               & graph H0 h0 : M"; 
-    proof (elim exE, intro exI conjI);
-      fix xi; 
+               & graph H0 h0 : M" 
+    proof (elim exE, intro exI conjI)
+      fix xi 
       assume a: "ALL y:H. - p (y + x0) - h y <= xi 
-                        & xi <= p (y + x0) - h y";
+                        & xi <= p (y + x0) - h y"
      
       txt {* Define $h_0$ as the canonical linear extension 
-      of $h$ on $H_0$:*};  
+      of $h$ on $H_0$:*}  
 
       def h0 ==
           "\\<lambda>x. let (y,a) = SOME (y, a). x = y + a ( * ) x0 & y:H
-               in (h y) + a * xi";
+               in (h y) + a * xi"
 
       txt {* We get that the graph of $h_0$ extends that of
-      $h$. *};
+      $h$. *}
         
-      have  "graph H h <= graph H0 h0"; 
-      proof (rule graph_extI);
-        fix t; assume "t:H"; 
-        have "(SOME (y, a). t = y + a ( * ) x0 & y : H) = (t,#0)";
-          by (rule decomp_H0_H, rule x0); 
-        thus "h t = h0 t"; by (simp! add: Let_def);
-      next;
-        show "H <= H0";
-        proof (rule subspace_subset);
-	  show "is_subspace H H0";
-          proof (unfold H0_def, rule subspace_vs_sum1);
-       	    show "is_vectorspace H"; ..;
-            show "is_vectorspace (lin x0)"; ..;
-          qed;
-        qed;
-      qed;
-      thus "g <= graph H0 h0"; by (simp!);
+      have  "graph H h <= graph H0 h0" 
+      proof (rule graph_extI)
+        fix t assume "t:H" 
+        have "(SOME (y, a). t = y + a ( * ) x0 & y : H) = (t,#0)"
+          by (rule decomp_H0_H, rule x0) 
+        thus "h t = h0 t" by (simp! add: Let_def)
+      next
+        show "H <= H0"
+        proof (rule subspace_subset)
+	  show "is_subspace H H0"
+          proof (unfold H0_def, rule subspace_vs_sum1)
+       	    show "is_vectorspace H" ..
+            show "is_vectorspace (lin x0)" ..
+          qed
+        qed
+      qed
+      thus "g <= graph H0 h0" by (simp!)
 
-      txt {* Apparently $h_0$ is not equal to $h$. *};
+      txt {* Apparently $h_0$ is not equal to $h$. *}
 
-      have "graph H h ~= graph H0 h0";
-      proof;
-        assume e: "graph H h = graph H0 h0";
-        have "x0 : H0"; 
-        proof (unfold H0_def, rule vs_sumI);
-          show "x0 = 00 + x0"; by (simp!);
-          from h; show "00 : H"; ..;
-          show "x0 : lin x0"; by (rule x_lin_x);
-        qed;
-        hence "(x0, h0 x0) : graph H0 h0"; ..;
-        with e; have "(x0, h0 x0) : graph H h"; by simp;
-        hence "x0 : H"; ..;
-        thus False; by contradiction;
-      qed;
-      thus "g ~= graph H0 h0"; by (simp!);
+      have "graph H h ~= graph H0 h0"
+      proof
+        assume e: "graph H h = graph H0 h0"
+        have "x0 : H0" 
+        proof (unfold H0_def, rule vs_sumI)
+          show "x0 = 00 + x0" by (simp!)
+          from h show "00 : H" ..
+          show "x0 : lin x0" by (rule x_lin_x)
+        qed
+        hence "(x0, h0 x0) : graph H0 h0" ..
+        with e have "(x0, h0 x0) : graph H h" by simp
+        hence "x0 : H" ..
+        thus False by contradiction
+      qed
+      thus "g ~= graph H0 h0" by (simp!)
 
       txt {* Furthermore  $h_0$ is a norm-preserving extension 
-     of $f$. *};
+     of $f$. *}
 
-      have "graph H0 h0 : norm_pres_extensions E p F f";
-      proof (rule norm_pres_extensionI2);
-        show "is_linearform H0 h0";
-          by (rule h0_lf, rule x0) (simp!)+;
-        show "is_subspace H0 E";
+      have "graph H0 h0 : norm_pres_extensions E p F f"
+      proof (rule norm_pres_extensionI2)
+        show "is_linearform H0 h0"
+          by (rule h0_lf, rule x0) (simp!)+
+        show "is_subspace H0 E"
           by (unfold H0_def, rule vs_sum_subspace, 
-             rule lin_subspace);
+             rule lin_subspace)
 
-        have "is_subspace F H"; .;
-        also; from h lin_vs; 
-	have [fold H0_def]: "is_subspace H (H + lin x0)"; ..;
-        finally (subspace_trans [OF _ h]); 
-	show f_h0: "is_subspace F H0"; .; (*** 
+        have "is_subspace F H" .
+        also from h lin_vs 
+	have [fold H0_def]: "is_subspace H (H + lin x0)" ..
+        finally (subspace_trans [OF _ h]) 
+	show f_h0: "is_subspace F H0" . (*** 
         backwards:
         show f_h0: "is_subspace F H0"; .;
         proof (rule subspace_trans [of F H H0]);
@@ -537,63 +537,63 @@
           thus "is_subspace H H0"; by (unfold H0_def);
         qed; ***)
 
-        show "graph F f <= graph H0 h0";
-        proof (rule graph_extI);
-          fix x; assume "x:F";
-          have "f x = h x"; ..;
-          also; have " ... = h x + #0 * xi"; by simp;
-          also; have "... = (let (y,a) = (x, #0) in h y + a * xi)";
-            by (simp add: Let_def);
-          also; have 
-            "(x, #0) = (SOME (y, a). x = y + a ( * ) x0 & y : H)";
-            by (rule decomp_H0_H [RS sym], rule x0) (simp!)+;
-          also; have 
+        show "graph F f <= graph H0 h0"
+        proof (rule graph_extI)
+          fix x assume "x:F"
+          have "f x = h x" ..
+          also have " ... = h x + #0 * xi" by simp
+          also have "... = (let (y,a) = (x, #0) in h y + a * xi)"
+            by (simp add: Let_def)
+          also have 
+            "(x, #0) = (SOME (y, a). x = y + a ( * ) x0 & y : H)"
+            by (rule decomp_H0_H [RS sym], rule x0) (simp!)+
+          also have 
             "(let (y,a) = (SOME (y,a). x = y + a ( * ) x0 & y : H)
               in h y + a * xi) 
-             = h0 x"; by (simp!);
-          finally; show "f x = h0 x"; .;
-        next;
-          from f_h0; show "F <= H0"; ..;
-        qed;
+             = h0 x" by (simp!)
+          finally show "f x = h0 x" .
+        next
+          from f_h0 show "F <= H0" ..
+        qed
 
-        show "ALL x:H0. h0 x <= p x";
-          by (rule h0_norm_pres, rule x0) (assumption | simp!)+;
-      qed;
-      thus "graph H0 h0 : M"; by (simp!);
-    qed;
-    thus ?thesis; ..;
-  qed;
+        show "ALL x:H0. h0 x <= p x"
+          by (rule h0_norm_pres, rule x0) (assumption | simp!)+
+      qed
+      thus "graph H0 h0 : M" by (simp!)
+    qed
+    thus ?thesis ..
+  qed
 
   txt {* We have shown that $h$ can still be extended to 
   some $h_0$, in contradiction to the assumption that 
-  $h$ is a maximal element. *};
+  $h$ is a maximal element. *}
 
-  hence "EX x:M. g <= x & g ~= x"; 
-    by (elim exE conjE, intro bexI conjI);
-  hence "~ (ALL x:M. g <= x --> g = x)"; by simp;
-  thus ?thesis; by contradiction;
-qed; 
+  hence "EX x:M. g <= x & g ~= x" 
+    by (elim exE conjE, intro bexI conjI)
+  hence "~ (ALL x:M. g <= x --> g = x)" by simp
+  thus ?thesis by contradiction
+qed 
 
-txt{* It follows $H = E$, and the thesis can be shown. *};
+txt{* It follows $H = E$, and the thesis can be shown. *}
 
 show "is_linearform E h & (ALL x:F. h x = f x) 
-     & (ALL x:E. h x <= p x)";
-proof (intro conjI); 
-  from eq; show "is_linearform E h"; by (simp!);
-  show "ALL x:F. h x = f x"; 
-  proof (intro ballI, rule sym);
-    fix x; assume "x:F"; show "f x = h x "; ..;
-  qed;
-  from eq; show "ALL x:E. h x <= p x"; by (force!);
-qed;
+     & (ALL x:E. h x <= p x)"
+proof (intro conjI) 
+  from eq show "is_linearform E h" by (simp!)
+  show "ALL x:F. h x = f x" 
+  proof (intro ballI, rule sym)
+    fix x assume "x:F" show "f x = h x " ..
+  qed
+  from eq show "ALL x:E. h x <= p x" by (force!)
+qed
 
-qed;
-qed;
-qed;
+qed
+qed
+qed
 *)
 
 
-subsection  {* Alternative formulation *};
+subsection  {* Alternative formulation *}
 
 text {* The following alternative formulation of the Hahn-Banach
 Theorem\label{abs-HahnBanach} uses the fact that for a real linear form
@@ -604,35 +604,35 @@
 \forall x\in H.\ap |h\ap x|\leq p\ap x& {\rm and}\\
 \forall x\in H.\ap h\ap x\leq p\ap x\\
 \end{matharray}
-*};
+*}
 
 theorem abs_HahnBanach:
   "[| is_vectorspace E; is_subspace F E; is_linearform F f; 
   is_seminorm E p; ALL x:F. abs (f x) <= p x |]
   ==> EX g. is_linearform E g & (ALL x:F. g x = f x)
-   & (ALL x:E. abs (g x) <= p x)";
-proof -;
+   & (ALL x:E. abs (g x) <= p x)"
+proof -
   assume "is_vectorspace E" "is_subspace F E" "is_seminorm E p" 
-    "is_linearform F f"  "ALL x:F. abs (f x) <= p x";
-  have "ALL x:F. f x <= p x";  by (rule abs_ineq_iff [RS iffD1]);
+    "is_linearform F f"  "ALL x:F. abs (f x) <= p x"
+  have "ALL x:F. f x <= p x"  by (rule abs_ineq_iff [RS iffD1])
   hence "EX g. is_linearform E g & (ALL x:F. g x = f x) 
-              & (ALL x:E. g x <= p x)";
-    by (simp! only: HahnBanach);
-  thus ?thesis; 
-  proof (elim exE conjE);
-    fix g; assume "is_linearform E g" "ALL x:F. g x = f x" 
-                  "ALL x:E. g x <= p x";
-    hence "ALL x:E. abs (g x) <= p x";
-      by (simp! add: abs_ineq_iff [OF subspace_refl]);
-    thus ?thesis; by (intro exI conjI);
-  qed;
-qed;
+              & (ALL x:E. g x <= p x)"
+    by (simp! only: HahnBanach)
+  thus ?thesis 
+  proof (elim exE conjE)
+    fix g assume "is_linearform E g" "ALL x:F. g x = f x" 
+                  "ALL x:E. g x <= p x"
+    hence "ALL x:E. abs (g x) <= p x"
+      by (simp! add: abs_ineq_iff [OF subspace_refl])
+    thus ?thesis by (intro exI conjI)
+  qed
+qed
 
-subsection {* The Hahn-Banach Theorem for normed spaces *};
+subsection {* The Hahn-Banach Theorem for normed spaces *}
 
 text {* Every continuous linear form $f$ on a subspace $F$ of a
 norm space $E$, can be extended to a continuous linear form $g$ on
-$E$ such that $\fnorm{f} = \fnorm {g}$. *};
+$E$ such that $\fnorm{f} = \fnorm {g}$. *}
 
 theorem norm_HahnBanach:
   "[| is_normed_vectorspace E norm; is_subspace F E; 
@@ -640,119 +640,119 @@
   ==> EX g. is_linearform E g
          & is_continuous E norm g 
          & (ALL x:F. g x = f x) 
-         & function_norm E norm g = function_norm F norm f";
-proof -;
-  assume e_norm: "is_normed_vectorspace E norm";
-  assume f: "is_subspace F E" "is_linearform F f";
-  assume f_cont: "is_continuous F norm f";
-  have e: "is_vectorspace E"; ..;
-  with _; have f_norm: "is_normed_vectorspace F norm"; ..;
+         & function_norm E norm g = function_norm F norm f"
+proof -
+  assume e_norm: "is_normed_vectorspace E norm"
+  assume f: "is_subspace F E" "is_linearform F f"
+  assume f_cont: "is_continuous F norm f"
+  have e: "is_vectorspace E" ..
+  with _ have f_norm: "is_normed_vectorspace F norm" ..
 
   txt{* We define a function $p$ on $E$ as follows:
   \begin{matharray}{l}
   p \: x = \fnorm f \cdot \norm x\\
   \end{matharray}
-  *};
+  *}
 
-  def p == "\\<lambda>x. function_norm F norm f * norm x";
+  def p == "\\<lambda>x. function_norm F norm f * norm x"
   
-  txt{* $p$ is a seminorm on $E$: *};
+  txt{* $p$ is a seminorm on $E$: *}
 
-  have q: "is_seminorm E p";
-  proof;
-    fix x y a; assume "x:E" "y:E";
+  have q: "is_seminorm E p"
+  proof
+    fix x y a assume "x:E" "y:E"
 
-    txt{* $p$ is positive definite: *};
+    txt{* $p$ is positive definite: *}
 
-    show "#0 <= p x";
-    proof (unfold p_def, rule real_le_mult_order1a);
-      from _ f_norm; show "#0 <= function_norm F norm f"; ..;
-      show "#0 <= norm x"; ..;
-    qed;
+    show "#0 <= p x"
+    proof (unfold p_def, rule real_le_mult_order1a)
+      from _ f_norm show "#0 <= function_norm F norm f" ..
+      show "#0 <= norm x" ..
+    qed
 
-    txt{* $p$ is absolutely homogenous: *};
+    txt{* $p$ is absolutely homogenous: *}
 
-    show "p (a (*) x) = abs a * p x";
-    proof -; 
-      have "p (a (*) x) = function_norm F norm f * norm (a (*) x)";
-        by (simp!);
-      also; have "norm (a (*) x) = abs a * norm x"; 
-        by (rule normed_vs_norm_abs_homogenous);
-      also; have "function_norm F norm f * (abs a * norm x) 
-        = abs a * (function_norm F norm f * norm x)";
-        by (simp! only: real_mult_left_commute);
-      also; have "... = abs a * p x"; by (simp!);
-      finally; show ?thesis; .;
-    qed;
+    show "p (a (*) x) = abs a * p x"
+    proof - 
+      have "p (a (*) x) = function_norm F norm f * norm (a (*) x)"
+        by (simp!)
+      also have "norm (a (*) x) = abs a * norm x" 
+        by (rule normed_vs_norm_abs_homogenous)
+      also have "function_norm F norm f * (abs a * norm x) 
+        = abs a * (function_norm F norm f * norm x)"
+        by (simp! only: real_mult_left_commute)
+      also have "... = abs a * p x" by (simp!)
+      finally show ?thesis .
+    qed
 
-    txt{* Furthermore, $p$ is subadditive: *};
+    txt{* Furthermore, $p$ is subadditive: *}
 
-    show "p (x + y) <= p x + p y";
-    proof -;
-      have "p (x + y) = function_norm F norm f * norm (x + y)";
-        by (simp!);
-      also; 
-      have "... <= function_norm F norm f * (norm x + norm y)";
-      proof (rule real_mult_le_le_mono1a);
-        from _ f_norm; show "#0 <= function_norm F norm f"; ..;
-        show "norm (x + y) <= norm x + norm y"; ..;
-      qed;
-      also; have "... = function_norm F norm f * norm x 
-                        + function_norm F norm f * norm y";
-        by (simp! only: real_add_mult_distrib2);
-      finally; show ?thesis; by (simp!);
-    qed;
-  qed;
+    show "p (x + y) <= p x + p y"
+    proof -
+      have "p (x + y) = function_norm F norm f * norm (x + y)"
+        by (simp!)
+      also 
+      have "... <= function_norm F norm f * (norm x + norm y)"
+      proof (rule real_mult_le_le_mono1a)
+        from _ f_norm show "#0 <= function_norm F norm f" ..
+        show "norm (x + y) <= norm x + norm y" ..
+      qed
+      also have "... = function_norm F norm f * norm x 
+                        + function_norm F norm f * norm y"
+        by (simp! only: real_add_mult_distrib2)
+      finally show ?thesis by (simp!)
+    qed
+  qed
 
-  txt{* $f$ is bounded by $p$. *}; 
+  txt{* $f$ is bounded by $p$. *} 
 
-  have "ALL x:F. abs (f x) <= p x";
-  proof;
-    fix x; assume "x:F";
-     from f_norm; show "abs (f x) <= p x"; 
-       by (simp! add: norm_fx_le_norm_f_norm_x);
-  qed;
+  have "ALL x:F. abs (f x) <= p x"
+  proof
+    fix x assume "x:F"
+     from f_norm show "abs (f x) <= p x" 
+       by (simp! add: norm_fx_le_norm_f_norm_x)
+  qed
 
   txt{* Using the fact that $p$ is a seminorm and 
   $f$ is bounded by $p$ we can apply the Hahn-Banach Theorem 
   for real vector spaces. 
   So $f$ can be extended in a norm-preserving way to some function
-  $g$ on the whole vector space $E$. *};
+  $g$ on the whole vector space $E$. *}
 
-  with e f q; 
+  with e f q 
   have "EX g. is_linearform E g & (ALL x:F. g x = f x) 
-            & (ALL x:E. abs (g x) <= p x)";
-    by (simp! add: abs_HahnBanach);
+            & (ALL x:E. abs (g x) <= p x)"
+    by (simp! add: abs_HahnBanach)
 
-  thus ?thesis;
-  proof (elim exE conjE); 
-    fix g;
+  thus ?thesis
+  proof (elim exE conjE) 
+    fix g
     assume "is_linearform E g" and a: "ALL x:F. g x = f x" 
-       and b: "ALL x:E. abs (g x) <= p x";
+       and b: "ALL x:E. abs (g x) <= p x"
 
     show "EX g. is_linearform E g 
             & is_continuous E norm g 
             & (ALL x:F. g x = f x) 
-            & function_norm E norm g = function_norm F norm f";
-    proof (intro exI conjI);
+            & function_norm E norm g = function_norm F norm f"
+    proof (intro exI conjI)
 
     txt{* We furthermore have to show that 
-    $g$ is also continuous: *};
+    $g$ is also continuous: *}
 
-      show g_cont: "is_continuous E norm g";
-      proof;
-        fix x; assume "x:E";
-        from b [RS bspec, OF this]; 
-        show "abs (g x) <= function_norm F norm f * norm x";
-          by (unfold p_def);
-      qed; 
+      show g_cont: "is_continuous E norm g"
+      proof
+        fix x assume "x:E"
+        from b [RS bspec, OF this] 
+        show "abs (g x) <= function_norm F norm f * norm x"
+          by (unfold p_def)
+      qed 
 
       txt {* To complete the proof, we show that 
-      $\fnorm g = \fnorm f$. \label{order_antisym} *};
+      $\fnorm g = \fnorm f$. \label{order_antisym} *}
 
       show "function_norm E norm g = function_norm F norm f"
-        (is "?L = ?R");
-      proof (rule order_antisym);
+        (is "?L = ?R")
+      proof (rule order_antisym)
 
         txt{* First we show $\fnorm g \leq \fnorm f$.  The function norm
         $\fnorm g$ is defined as the smallest $c\in\bbbR$ such that
@@ -763,42 +763,42 @@
         \begin{matharray}{l}
         \All {x\in E} {|g\ap x| \leq \fnorm f \cdot \norm x}
         \end{matharray}
-        *};
+        *}
  
-        have "ALL x:E. abs (g x) <= function_norm F norm f * norm x";
-        proof;
-          fix x; assume "x:E"; 
-          show "abs (g x) <= function_norm F norm f * norm x";
-            by (simp!);
-        qed;
+        have "ALL x:E. abs (g x) <= function_norm F norm f * norm x"
+        proof
+          fix x assume "x:E" 
+          show "abs (g x) <= function_norm F norm f * norm x"
+            by (simp!)
+        qed
 
-        with _ g_cont; show "?L <= ?R";
-        proof (rule fnorm_le_ub);
-          from f_cont f_norm; show "#0 <= function_norm F norm f"; ..;
-        qed;
+        with _ g_cont show "?L <= ?R"
+        proof (rule fnorm_le_ub)
+          from f_cont f_norm show "#0 <= function_norm F norm f" ..
+        qed
 
         txt{* The other direction is achieved by a similar 
-        argument. *};
+        argument. *}
 
-        have "ALL x:F. abs (f x) <= function_norm E norm g * norm x";
-        proof;
-          fix x; assume "x : F"; 
-          from a; have "g x = f x"; ..;
-          hence "abs (f x) = abs (g x)"; by simp;
-          also; from _ _ g_cont;
-          have "... <= function_norm E norm g * norm x";
-          proof (rule norm_fx_le_norm_f_norm_x);
-            show "x:E"; ..;
-          qed;
-          finally; show "abs (f x) <= function_norm E norm g * norm x"; .;
-        qed; 
-        thus "?R <= ?L"; 
-        proof (rule fnorm_le_ub [OF f_norm f_cont]);
-          from g_cont; show "#0 <= function_norm E norm g"; ..;
-        qed;
-      qed;
-    qed;
-  qed;
-qed;
+        have "ALL x:F. abs (f x) <= function_norm E norm g * norm x"
+        proof
+          fix x assume "x : F" 
+          from a have "g x = f x" ..
+          hence "abs (f x) = abs (g x)" by simp
+          also from _ _ g_cont
+          have "... <= function_norm E norm g * norm x"
+          proof (rule norm_fx_le_norm_f_norm_x)
+            show "x:E" ..
+          qed
+          finally show "abs (f x) <= function_norm E norm g * norm x" .
+        qed 
+        thus "?R <= ?L" 
+        proof (rule fnorm_le_ub [OF f_norm f_cont])
+          from g_cont show "#0 <= function_norm E norm g" ..
+        qed
+      qed
+    qed
+  qed
+qed
 
-end;
+end
\ No newline at end of file




