
diff -r ea4dc7603f0b -r 371f023d3dbd src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy
--- a/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy	Sun Jun 04 00:09:04 2000 +0200
+++ b/src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy	Sun Jun 04 19:39:29 2000 +0200
@@ -3,9 +3,9 @@
     Author:     Gertrud Bauer, TU Munich
 *)
 
-header {* Extending non-maximal functions *};
+header {* Extending non-maximal functions *}
 
-theory HahnBanachExtLemmas = FunctionNorm:;
+theory HahnBanachExtLemmas = FunctionNorm:
 
 text{* In this section the following context is presumed.
 Let $E$ be a real vector space with a 
@@ -19,7 +19,7 @@
 $h_0\ap x = h\ap y + a \cdot \xi$ for a certain $\xi$.
 
 Subsequently we show some properties of this extension $h_0$ of $h$.
-*}; 
+*} 
 
 
 text {* This lemma will be used to show the existence of a linear
@@ -32,212 +32,212 @@
 it suffices to show that 
 \begin{matharray}{l} \All
 {u\in F}{\All {v\in F}{a\ap u \leq b \ap v}} 
-\end{matharray} *};
+\end{matharray} *}
 
 lemma ex_xi: 
   "[| is_vectorspace F; !! u v. [| u:F; v:F |] ==> a u <= b v |]
-  ==> EX (xi::real). ALL y:F. a y <= xi & xi <= b y"; 
-proof -;
-  assume vs: "is_vectorspace F";
-  assume r: "(!! u v. [| u:F; v:F |] ==> a u <= (b v::real))";
+  ==> EX (xi::real). ALL y:F. a y <= xi & xi <= b y" 
+proof -
+  assume vs: "is_vectorspace F"
+  assume r: "(!! u v. [| u:F; v:F |] ==> a u <= (b v::real))"
 
   txt {* From the completeness of the reals follows:
   The set $S = \{a\: u\dt\: u\in F\}$ has a supremum, if
-  it is non-empty and has an upper bound. *};
+  it is non-empty and has an upper bound. *}
 
-  let ?S = "{a u :: real | u. u:F}";
+  let ?S = "{a u :: real | u. u:F}"
 
-  have "EX xi. isLub UNIV ?S xi";  
-  proof (rule reals_complete);
+  have "EX xi. isLub UNIV ?S xi"  
+  proof (rule reals_complete)
   
-    txt {* The set $S$ is non-empty, since $a\ap\zero \in S$: *};
+    txt {* The set $S$ is non-empty, since $a\ap\zero \in S$: *}
 
-    from vs; have "a 00 : ?S"; by force;
-    thus "EX X. X : ?S"; ..;
+    from vs have "a 00 : ?S" by force
+    thus "EX X. X : ?S" ..
 
-    txt {* $b\ap \zero$ is an upper bound of $S$: *};
+    txt {* $b\ap \zero$ is an upper bound of $S$: *}
 
-    show "EX Y. isUb UNIV ?S Y"; 
-    proof; 
-      show "isUb UNIV ?S (b 00)";
-      proof (intro isUbI setleI ballI);
-        show "b 00 : UNIV"; ..;
-      next;
+    show "EX Y. isUb UNIV ?S Y" 
+    proof 
+      show "isUb UNIV ?S (b 00)"
+      proof (intro isUbI setleI ballI)
+        show "b 00 : UNIV" ..
+      next
 
-        txt {* Every element $y\in S$ is less than $b\ap \zero$: *};
+        txt {* Every element $y\in S$ is less than $b\ap \zero$: *}
 
-        fix y; assume y: "y : ?S"; 
-        from y; have "EX u:F. y = a u"; by fast;
-        thus "y <= b 00"; 
-        proof;
-          fix u; assume "u:F"; 
-          assume "y = a u";
-          also; have "a u <= b 00"; by (rule r) (simp!)+;
-          finally; show ?thesis; .;
-        qed;
-      qed;
-    qed;
-  qed;
+        fix y assume y: "y : ?S" 
+        from y have "EX u:F. y = a u" by fast
+        thus "y <= b 00" 
+        proof
+          fix u assume "u:F" 
+          assume "y = a u"
+          also have "a u <= b 00" by (rule r) (simp!)+
+          finally show ?thesis .
+        qed
+      qed
+    qed
+  qed
 
-  thus "EX xi. ALL y:F. a y <= xi & xi <= b y"; 
-  proof (elim exE);
-    fix xi; assume "isLub UNIV ?S xi"; 
-    show ?thesis;
-    proof (intro exI conjI ballI); 
+  thus "EX xi. ALL y:F. a y <= xi & xi <= b y" 
+  proof (elim exE)
+    fix xi assume "isLub UNIV ?S xi" 
+    show ?thesis
+    proof (intro exI conjI ballI) 
    
-      txt {* For all $y\in F$ holds $a\ap y \leq \xi$: *};
+      txt {* For all $y\in F$ holds $a\ap y \leq \xi$: *}
      
-      fix y; assume y: "y:F";
-      show "a y <= xi";    
-      proof (rule isUbD);  
-        show "isUb UNIV ?S xi"; ..;
-      qed (force!);
-    next;
+      fix y assume y: "y:F"
+      show "a y <= xi"    
+      proof (rule isUbD)  
+        show "isUb UNIV ?S xi" ..
+      qed (force!)
+    next
 
-      txt {* For all $y\in F$ holds $\xi\leq b\ap y$: *};
+      txt {* For all $y\in F$ holds $\xi\leq b\ap y$: *}
 
-      fix y; assume "y:F";
-      show "xi <= b y";  
-      proof (intro isLub_le_isUb isUbI setleI);
-        show "b y : UNIV"; ..;
-        show "ALL ya : ?S. ya <= b y"; 
-        proof;
-          fix au; assume au: "au : ?S ";
-          hence "EX u:F. au = a u"; by fast;
-          thus "au <= b y";
-          proof;
-            fix u; assume "u:F"; assume "au = a u";  
-            also; have "... <= b y"; by (rule r);
-            finally; show ?thesis; .;
-          qed;
-        qed;
-      qed; 
-    qed;
-  qed;
-qed;
+      fix y assume "y:F"
+      show "xi <= b y"  
+      proof (intro isLub_le_isUb isUbI setleI)
+        show "b y : UNIV" ..
+        show "ALL ya : ?S. ya <= b y" 
+        proof
+          fix au assume au: "au : ?S "
+          hence "EX u:F. au = a u" by fast
+          thus "au <= b y"
+          proof
+            fix u assume "u:F" assume "au = a u"  
+            also have "... <= b y" by (rule r)
+            finally show ?thesis .
+          qed
+        qed
+      qed 
+    qed
+  qed
+qed
 
 text{* \medskip The function $h_0$ is defined as a
 $h_0\ap x = h\ap y + a\cdot \xi$ where $x = y + a\mult \xi$
-is a linear extension of $h$ to $H_0$. *};
+is a linear extension of $h$ to $H_0$. *}
 
 lemma h0_lf: 
   "[| h0 == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a (*) x0 & y:H 
                 in h y + a * xi);
   H0 == H + lin x0; is_subspace H E; is_linearform H h; x0 ~: H; 
   x0 : E; x0 ~= 00; is_vectorspace E |]
-  ==> is_linearform H0 h0";
-proof -;
+  ==> is_linearform H0 h0"
+proof -
   assume h0_def: 
     "h0 == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a (*) x0 & y:H 
                in h y + a * xi)"
     and H0_def: "H0 == H + lin x0" 
     and vs: "is_subspace H E" "is_linearform H h" "x0 ~: H"
-      "x0 ~= 00" "x0 : E" "is_vectorspace E";
+      "x0 ~= 00" "x0 : E" "is_vectorspace E"
 
-  have h0: "is_vectorspace H0"; 
-  proof (unfold H0_def, rule vs_sum_vs);
-    show "is_subspace (lin x0) E"; ..;
-  qed; 
+  have h0: "is_vectorspace H0" 
+  proof (unfold H0_def, rule vs_sum_vs)
+    show "is_subspace (lin x0) E" ..
+  qed 
 
-  show ?thesis;
-  proof;
-    fix x1 x2; assume x1: "x1 : H0" and x2: "x2 : H0"; 
+  show ?thesis
+  proof
+    fix x1 x2 assume x1: "x1 : H0" and x2: "x2 : H0" 
 
     txt{* We now have to show that $h_0$ is additive, i.~e.\
     $h_0 \ap (x_1\plus x_2) = h_0\ap x_1 + h_0\ap x_2$
-    for $x_1, x_2\in H$. *}; 
+    for $x_1, x_2\in H$. *} 
 
-    have x1x2: "x1 + x2 : H0"; 
-      by (rule vs_add_closed, rule h0); 
-    from x1; 
-    have ex_x1: "EX y1 a1. x1 = y1 + a1 (*) x0  & y1 : H"; 
-      by (unfold H0_def vs_sum_def lin_def) fast;
-    from x2; 
-    have ex_x2: "EX y2 a2. x2 = y2 + a2 (*) x0 & y2 : H"; 
-      by (unfold H0_def vs_sum_def lin_def) fast;
-    from x1x2; 
-    have ex_x1x2: "EX y a. x1 + x2 = y + a (*) x0 & y : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
+    have x1x2: "x1 + x2 : H0" 
+      by (rule vs_add_closed, rule h0) 
+    from x1 
+    have ex_x1: "EX y1 a1. x1 = y1 + a1 (*) x0  & y1 : H" 
+      by (unfold H0_def vs_sum_def lin_def) fast
+    from x2 
+    have ex_x2: "EX y2 a2. x2 = y2 + a2 (*) x0 & y2 : H" 
+      by (unfold H0_def vs_sum_def lin_def) fast
+    from x1x2 
+    have ex_x1x2: "EX y a. x1 + x2 = y + a (*) x0 & y : H"
+      by (unfold H0_def vs_sum_def lin_def) fast
 
-    from ex_x1 ex_x2 ex_x1x2;
-    show "h0 (x1 + x2) = h0 x1 + h0 x2";
-    proof (elim exE conjE);
-      fix y1 y2 y a1 a2 a;
+    from ex_x1 ex_x2 ex_x1x2
+    show "h0 (x1 + x2) = h0 x1 + h0 x2"
+    proof (elim exE conjE)
+      fix y1 y2 y a1 a2 a
       assume y1: "x1 = y1 + a1 (*) x0"     and y1': "y1 : H"
          and y2: "x2 = y2 + a2 (*) x0"     and y2': "y2 : H" 
-         and y: "x1 + x2 = y + a (*) x0"   and y':  "y  : H"; 
+         and y: "x1 + x2 = y + a (*) x0"   and y':  "y  : H" 
 
-      have ya: "y1 + y2 = y & a1 + a2 = a"; 
-      proof (rule decomp_H0);;
-	txt_raw {* \label{decomp-H0-use} *};;
-        show "y1 + y2 + (a1 + a2) (*) x0 = y + a (*) x0"; 
-          by (simp! add: vs_add_mult_distrib2 [of E]);
-        show "y1 + y2 : H"; ..;
-      qed;
+      have ya: "y1 + y2 = y & a1 + a2 = a" 
+      proof (rule decomp_H0)
+	txt_raw {* \label{decomp-H0-use} *}
+        show "y1 + y2 + (a1 + a2) (*) x0 = y + a (*) x0" 
+          by (simp! add: vs_add_mult_distrib2 [of E])
+        show "y1 + y2 : H" ..
+      qed
 
-      have "h0 (x1 + x2) = h y + a * xi";
-	by (rule h0_definite);
-      also; have "... = h (y1 + y2) + (a1 + a2) * xi"; 
-        by (simp add: ya);
-      also; from vs y1' y2'; 
-      have "... = h y1 + h y2 + a1 * xi + a2 * xi"; 
+      have "h0 (x1 + x2) = h y + a * xi"
+	by (rule h0_definite)
+      also have "... = h (y1 + y2) + (a1 + a2) * xi" 
+        by (simp add: ya)
+      also from vs y1' y2' 
+      have "... = h y1 + h y2 + a1 * xi + a2 * xi" 
 	by (simp add: linearform_add [of H] 
-                      real_add_mult_distrib);
-      also; have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"; 
-        by simp;
-      also; have "h y1 + a1 * xi = h0 x1"; 
-        by (rule h0_definite [RS sym]);
-      also; have "h y2 + a2 * xi = h0 x2"; 
-        by (rule h0_definite [RS sym]);
-      finally; show ?thesis; .;
-    qed;
+                      real_add_mult_distrib)
+      also have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)" 
+        by simp
+      also have "h y1 + a1 * xi = h0 x1" 
+        by (rule h0_definite [RS sym])
+      also have "h y2 + a2 * xi = h0 x2" 
+        by (rule h0_definite [RS sym])
+      finally show ?thesis .
+    qed
  
     txt{* We further have to show that $h_0$ is multiplicative, 
     i.~e.\ $h_0\ap (c \mult x_1) = c \cdot h_0\ap x_1$
     for $x\in H$ and $c\in \bbbR$. 
-    *}; 
+    *} 
 
-  next;  
-    fix c x1; assume x1: "x1 : H0";    
-    have ax1: "c (*) x1 : H0";
-      by (rule vs_mult_closed, rule h0);
-    from x1; have ex_x: "!! x. x: H0 
-                        ==> EX y a. x = y + a (*) x0 & y : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
+  next  
+    fix c x1 assume x1: "x1 : H0"    
+    have ax1: "c (*) x1 : H0"
+      by (rule vs_mult_closed, rule h0)
+    from x1 have ex_x: "!! x. x: H0 
+                        ==> EX y a. x = y + a (*) x0 & y : H"
+      by (unfold H0_def vs_sum_def lin_def) fast
 
-    from x1; have ex_x1: "EX y1 a1. x1 = y1 + a1 (*) x0 & y1 : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
-    with ex_x [of "c (*) x1", OF ax1];
-    show "h0 (c (*) x1) = c * (h0 x1)";  
-    proof (elim exE conjE);
-      fix y1 y a1 a; 
+    from x1 have ex_x1: "EX y1 a1. x1 = y1 + a1 (*) x0 & y1 : H"
+      by (unfold H0_def vs_sum_def lin_def) fast
+    with ex_x [of "c (*) x1", OF ax1]
+    show "h0 (c (*) x1) = c * (h0 x1)"  
+    proof (elim exE conjE)
+      fix y1 y a1 a 
       assume y1: "x1 = y1 + a1 (*) x0"       and y1': "y1 : H"
-        and y: "c (*) x1 = y  + a  (*) x0"   and y':  "y  : H"; 
+        and y: "c (*) x1 = y  + a  (*) x0"   and y':  "y  : H" 
 
-      have ya: "c (*) y1 = y & c * a1 = a"; 
-      proof (rule decomp_H0); 
-	show "c (*) y1 + (c * a1) (*) x0 = y + a (*) x0"; 
-          by (simp! add: add: vs_add_mult_distrib1);
-        show "c (*) y1 : H"; ..;
-      qed;
+      have ya: "c (*) y1 = y & c * a1 = a" 
+      proof (rule decomp_H0) 
+	show "c (*) y1 + (c * a1) (*) x0 = y + a (*) x0" 
+          by (simp! add: add: vs_add_mult_distrib1)
+        show "c (*) y1 : H" ..
+      qed
 
-      have "h0 (c (*) x1) = h y + a * xi"; 
-	by (rule h0_definite);
-      also; have "... = h (c (*) y1) + (c * a1) * xi";
-        by (simp add: ya);
-      also; from vs y1'; have "... = c * h y1 + c * a1 * xi"; 
-	by (simp add: linearform_mult [of H]);
-      also; from vs y1'; have "... = c * (h y1 + a1 * xi)"; 
-	by (simp add: real_add_mult_distrib2 real_mult_assoc);
-      also; have "h y1 + a1 * xi = h0 x1"; 
-        by (rule h0_definite [RS sym]);
-      finally; show ?thesis; .;
-    qed;
-  qed;
-qed;
+      have "h0 (c (*) x1) = h y + a * xi" 
+	by (rule h0_definite)
+      also have "... = h (c (*) y1) + (c * a1) * xi"
+        by (simp add: ya)
+      also from vs y1' have "... = c * h y1 + c * a1 * xi" 
+	by (simp add: linearform_mult [of H])
+      also from vs y1' have "... = c * (h y1 + a1 * xi)" 
+	by (simp add: real_add_mult_distrib2 real_mult_assoc)
+      also have "h y1 + a1 * xi = h0 x1" 
+        by (rule h0_definite [RS sym])
+      finally show ?thesis .
+    qed
+  qed
+qed
 
 text{* \medskip The linear extension $h_0$ of $h$
-is bounded by the seminorm $p$. *};
+is bounded by the seminorm $p$. *}
 
 lemma h0_norm_pres:
   "[| h0 == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a (*) x0 & y:H 
@@ -245,105 +245,105 @@
   H0 == H + lin x0; x0 ~: H; x0 : E; x0 ~= 00; is_vectorspace E; 
   is_subspace H E; is_seminorm E p; is_linearform H h; ALL y:H. h y <= p y; 
   ALL y:H. - p (y + x0) - h y <= xi & xi <= p (y + x0) - h y |]
-   ==> ALL x:H0. h0 x <= p x"; 
-proof; 
+   ==> ALL x:H0. h0 x <= p x" 
+proof 
   assume h0_def: 
     "h0 == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a (*) x0 & y:H 
                in (h y) + a * xi)"
     and H0_def: "H0 == H + lin x0" 
     and vs: "x0 ~: H" "x0 : E" "x0 ~= 00" "is_vectorspace E" 
             "is_subspace H E" "is_seminorm E p" "is_linearform H h" 
-    and a: "ALL y:H. h y <= p y";
-  presume a1: "ALL ya:H. - p (ya + x0) - h ya <= xi";
-  presume a2: "ALL ya:H. xi <= p (ya + x0) - h ya";
-  fix x; assume "x : H0"; 
+    and a: "ALL y:H. h y <= p y"
+  presume a1: "ALL ya:H. - p (ya + x0) - h ya <= xi"
+  presume a2: "ALL ya:H. xi <= p (ya + x0) - h ya"
+  fix x assume "x : H0" 
   have ex_x: 
-    "!! x. x : H0 ==> EX y a. x = y + a (*) x0 & y : H";
-    by (unfold H0_def vs_sum_def lin_def) fast;
-  have "EX y a. x = y + a (*) x0 & y : H";
-    by (rule ex_x);
-  thus "h0 x <= p x";
-  proof (elim exE conjE);
-    fix y a; assume x: "x = y + a (*) x0" and y: "y : H";
-    have "h0 x = h y + a * xi";
-      by (rule h0_definite);
+    "!! x. x : H0 ==> EX y a. x = y + a (*) x0 & y : H"
+    by (unfold H0_def vs_sum_def lin_def) fast
+  have "EX y a. x = y + a (*) x0 & y : H"
+    by (rule ex_x)
+  thus "h0 x <= p x"
+  proof (elim exE conjE)
+    fix y a assume x: "x = y + a (*) x0" and y: "y : H"
+    have "h0 x = h y + a * xi"
+      by (rule h0_definite)
 
     txt{* Now we show  
     $h\ap y + a \cdot \xi\leq  p\ap (y\plus a \mult x_0)$ 
-    by case analysis on $a$. \label{linorder_linear_split}*};
+    by case analysis on $a$. \label{linorder_linear_split}*}
 
-    also; have "... <= p (y + a (*) x0)";
-    proof (rule linorder_linear_split); 
+    also have "... <= p (y + a (*) x0)"
+    proof (rule linorder_linear_split) 
 
-      assume z: "a = (#0::real)"; 
-      with vs y a; show ?thesis; by simp;
+      assume z: "a = #0"
+      with vs y a show ?thesis by simp
 
     txt {* In the case $a < 0$, we use $a_1$ with $\idt{ya}$ 
-    taken as $y/a$: *};
+    taken as $y/a$: *}
 
-    next;
-      assume lz: "a < #0"; hence nz: "a ~= #0"; by simp;
-      from a1; 
-      have "- p (rinv a (*) y + x0) - h (rinv a (*) y) <= xi";
-        by (rule bspec) (simp!);
+    next
+      assume lz: "a < #0" hence nz: "a ~= #0" by simp
+      from a1 
+      have "- p (rinv a (*) y + x0) - h (rinv a (*) y) <= xi"
+        by (rule bspec) (simp!)
 
       txt {* The thesis for this case now follows by a short  
-      calculation. *};      
+      calculation. *}      
       hence "a * xi 
-            <= a * (- p (rinv a (*) y + x0) - h (rinv a (*) y))";
-        by (rule real_mult_less_le_anti [OF lz]);
-      also; have "... = - a * (p (rinv a (*) y + x0)) 
-                        - a * (h (rinv a (*) y))";
-        by (rule real_mult_diff_distrib);
-      also; from lz vs y; have "- a * (p (rinv a (*) y + x0)) 
-                               = p (a (*) (rinv a (*) y + x0))";
-        by (simp add: seminorm_abs_homogenous abs_minus_eqI2);
-      also; from nz vs y; have "... = p (y + a (*) x0)";
-        by (simp add: vs_add_mult_distrib1);
-      also; from nz vs y; have "a * (h (rinv a (*) y)) =  h y";
-        by (simp add: linearform_mult [RS sym]);
-      finally; have "a * xi <= p (y + a (*) x0) - h y"; .;
+            <= a * (- p (rinv a (*) y + x0) - h (rinv a (*) y))"
+        by (rule real_mult_less_le_anti [OF lz])
+      also have "... = - a * (p (rinv a (*) y + x0)) 
+                        - a * (h (rinv a (*) y))"
+        by (rule real_mult_diff_distrib)
+      also from lz vs y have "- a * (p (rinv a (*) y + x0)) 
+                               = p (a (*) (rinv a (*) y + x0))"
+        by (simp add: seminorm_abs_homogenous abs_minus_eqI2)
+      also from nz vs y have "... = p (y + a (*) x0)"
+        by (simp add: vs_add_mult_distrib1)
+      also from nz vs y have "a * (h (rinv a (*) y)) =  h y"
+        by (simp add: linearform_mult [RS sym])
+      finally have "a * xi <= p (y + a (*) x0) - h y" .
 
-      hence "h y + a * xi <= h y + p (y + a (*) x0) - h y";
-        by (simp add: real_add_left_cancel_le);
-      thus ?thesis; by simp;
+      hence "h y + a * xi <= h y + p (y + a (*) x0) - h y"
+        by (simp add: real_add_left_cancel_le)
+      thus ?thesis by simp
 
       txt {* In the case $a > 0$, we use $a_2$ with $\idt{ya}$ 
-      taken as $y/a$: *};
+      taken as $y/a$: *}
 
-    next; 
-      assume gz: "#0 < a"; hence nz: "a ~= #0"; by simp;
-      from a2;
-      have "xi <= p (rinv a (*) y + x0) - h (rinv a (*) y)";
-        by (rule bspec) (simp!);
+    next 
+      assume gz: "#0 < a" hence nz: "a ~= #0" by simp
+      from a2
+      have "xi <= p (rinv a (*) y + x0) - h (rinv a (*) y)"
+        by (rule bspec) (simp!)
 
       txt {* The thesis for this case follows by a short
-      calculation: *};
+      calculation: *}
 
-      with gz; have "a * xi 
-            <= a * (p (rinv a (*) y + x0) - h (rinv a (*) y))";
-        by (rule real_mult_less_le_mono);
-      also; have "... = a * p (rinv a (*) y + x0) 
-                        - a * h (rinv a (*) y)";
-        by (rule real_mult_diff_distrib2); 
-      also; from gz vs y; 
+      with gz have "a * xi 
+            <= a * (p (rinv a (*) y + x0) - h (rinv a (*) y))"
+        by (rule real_mult_less_le_mono)
+      also have "... = a * p (rinv a (*) y + x0) 
+                        - a * h (rinv a (*) y)"
+        by (rule real_mult_diff_distrib2) 
+      also from gz vs y 
       have "a * p (rinv a (*) y + x0) 
-           = p (a (*) (rinv a (*) y + x0))";
-        by (simp add: seminorm_abs_homogenous abs_eqI2);
-      also; from nz vs y; 
-      have "... = p (y + a (*) x0)";
-        by (simp add: vs_add_mult_distrib1);
-      also; from nz vs y; have "a * h (rinv a (*) y) = h y";
-        by (simp add: linearform_mult [RS sym]); 
-      finally; have "a * xi <= p (y + a (*) x0) - h y"; .;
+           = p (a (*) (rinv a (*) y + x0))"
+        by (simp add: seminorm_abs_homogenous abs_eqI2)
+      also from nz vs y 
+      have "... = p (y + a (*) x0)"
+        by (simp add: vs_add_mult_distrib1)
+      also from nz vs y have "a * h (rinv a (*) y) = h y"
+        by (simp add: linearform_mult [RS sym]) 
+      finally have "a * xi <= p (y + a (*) x0) - h y" .
  
-      hence "h y + a * xi <= h y + (p (y + a (*) x0) - h y)";
-        by (simp add: real_add_left_cancel_le);
-      thus ?thesis; by simp;
-    qed;
-    also; from x; have "... = p x"; by simp;
-    finally; show ?thesis; .;
-  qed;
-qed blast+; 
+      hence "h y + a * xi <= h y + (p (y + a (*) x0) - h y)"
+        by (simp add: real_add_left_cancel_le)
+      thus ?thesis by simp
+    qed
+    also from x have "... = p x" by simp
+    finally show ?thesis .
+  qed
+qed blast+ 
 
-end;
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+end
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