equal
deleted
inserted
replaced
5922 finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> \<le> e/4" . |
5922 finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> \<le> e/4" . |
5923 |
5923 |
5924 next |
5924 next |
5925 have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x)))) |
5925 have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x)))) |
5926 \<le> norm (\<Sum>j = 0..s. \<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))" |
5926 \<le> norm (\<Sum>j = 0..s. \<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))" |
5927 apply (subst sum_group) |
5927 apply (subst sum.group) |
5928 using s by (auto simp: sum_subtractf split_def p'(1)) |
5928 using s by (auto simp: sum_subtractf split_def p'(1)) |
5929 also have "\<dots> < e/2" |
5929 also have "\<dots> < e/2" |
5930 proof - |
5930 proof - |
5931 have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) |
5931 have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) |
5932 \<le> (\<Sum>i = 0..s. e/2 ^ (i + 2))" |
5932 \<le> (\<Sum>i = 0..s. e/2 ^ (i + 2))" |