equal
deleted
inserted
replaced
3583 (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow> |
3583 (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow> |
3584 (f has_integral I) (cbox a b)" (is "?lhs = ?rhs") |
3584 (f has_integral I) (cbox a b)" (is "?lhs = ?rhs") |
3585 proof |
3585 proof |
3586 assume ?lhs |
3586 assume ?lhs |
3587 from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close> |
3587 from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close> |
3588 show ?rhs by (simp add: field_simps vector_add_divide_simps) |
3588 show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps) |
3589 next |
3589 next |
3590 assume ?rhs |
3590 assume ?rhs |
3591 from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close> |
3591 from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close> |
3592 show ?lhs by simp |
3592 show ?lhs by simp |
3593 qed |
3593 qed |
4251 show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3" |
4251 show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3" |
4252 by (metis SUMEQ d1_fin norm_minus_cancel) |
4252 by (metis SUMEQ d1_fin norm_minus_cancel) |
4253 show "norm (?SUM p - integral {a..t} f) < e/3" |
4253 show "norm (?SUM p - integral {a..t} f) < e/3" |
4254 using d2_fin by blast |
4254 using d2_fin by blast |
4255 show "norm ((c - t) *\<^sub>R f c) < e/3" |
4255 show "norm ((c - t) *\<^sub>R f c) < e/3" |
4256 using w cwt \<open>t < c\<close> by (auto simp add: field_simps) |
4256 using w cwt \<open>t < c\<close> by simp (simp add: field_simps) |
4257 qed |
4257 qed |
4258 then show ?thesis by simp |
4258 then show ?thesis by simp |
4259 qed |
4259 qed |
4260 qed |
4260 qed |
4261 qed |
4261 qed |