src/HOL/Real/HahnBanach/ZornLemma.thy
 changeset 27612 d3eb431db035 parent 23378 1d138d6bb461 child 29234 60f7fb56f8cd
```     1.1 --- a/src/HOL/Real/HahnBanach/ZornLemma.thy	Tue Jul 15 16:50:09 2008 +0200
1.2 +++ b/src/HOL/Real/HahnBanach/ZornLemma.thy	Tue Jul 15 19:39:37 2008 +0200
1.3 @@ -5,7 +5,9 @@
1.4
1.5  header {* Zorn's Lemma *}
1.6
1.7 -theory ZornLemma imports Zorn begin
1.8 +theory ZornLemma
1.9 +imports Zorn
1.10 +begin
1.11
1.12  text {*
1.13    Zorn's Lemmas states: if every linear ordered subset of an ordered
1.14 @@ -23,8 +25,6 @@
1.15      and aS: "a \<in> S"
1.16    shows "\<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> z \<longrightarrow> y = z"
1.17  proof (rule Zorn_Lemma2)
1.18 -  txt_raw {* \footnote{See
1.19 -  \url{http://isabelle.in.tum.de/library/HOL/HOL-Complex/Zorn.html}} \isanewline *}
1.20    show "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>z \<in> c. z \<subseteq> y"
1.21    proof
1.22      fix c assume "c \<in> chain S"
1.23 @@ -32,22 +32,22 @@
1.24      proof cases
1.25
1.26        txt {* If @{text c} is an empty chain, then every element in
1.27 -      @{text S} is an upper bound of @{text c}. *}
1.28 +	@{text S} is an upper bound of @{text c}. *}
1.29
1.30        assume "c = {}"
1.31        with aS show ?thesis by fast
1.32
1.33        txt {* If @{text c} is non-empty, then @{text "\<Union>c"} is an upper
1.34 -      bound of @{text c}, lying in @{text S}. *}
1.35 +	bound of @{text c}, lying in @{text S}. *}
1.36
1.37      next
1.38 -      assume c: "c \<noteq> {}"
1.39 +      assume "c \<noteq> {}"
1.40        show ?thesis
1.41        proof
1.42          show "\<forall>z \<in> c. z \<subseteq> \<Union>c" by fast
1.43          show "\<Union>c \<in> S"
1.44          proof (rule r)
1.45 -          from c show "\<exists>x. x \<in> c" by fast
1.46 +          from `c \<noteq> {}` show "\<exists>x. x \<in> c" by fast
1.47  	  show "c \<in> chain S" by fact
1.48          qed
1.49        qed
```