--- a/src/HOL/Probability/Lebesgue_Measure.thy Thu Dec 02 14:57:21 2010 +0100
+++ b/src/HOL/Probability/Lebesgue_Measure.thy Thu Dec 02 14:57:50 2010 +0100
@@ -755,20 +755,6 @@
qed
qed
-lemma Real_mult_nonneg: assumes "x \<ge> 0" "y \<ge> 0"
- shows "Real (x * y) = Real x * Real y" using assms by auto
-
-lemma Real_setprod: assumes "\<forall>x\<in>A. f x \<ge> 0" shows "Real (setprod f A) = setprod (\<lambda>x. Real (f x)) A"
-proof(cases "finite A")
- case True thus ?thesis using assms
- proof(induct A) case (insert x A)
- have "0 \<le> setprod f A" apply(rule setprod_nonneg) using insert by auto
- thus ?case unfolding setprod_insert[OF insert(1-2)] apply-
- apply(subst Real_mult_nonneg) prefer 3 apply(subst insert(3)[THEN sym])
- using insert by auto
- qed auto
-qed auto
-
lemma e2p_Int:"e2p ` A \<inter> e2p ` B = e2p ` (A \<inter> B)" (is "?L = ?R")
apply(rule image_Int[THEN sym]) using bij_euclidean_component
unfolding bij_betw_def by auto