diff -r c34b072518c9 -r 6f09346c7c08 src/HOL/Real.thy
--- a/src/HOL/Real.thy	Mon Oct 05 16:41:06 2009 +0100
+++ b/src/HOL/Real.thy	Mon Oct 05 17:27:46 2009 +0100
@@ -2,4 +2,28 @@
 imports RComplete RealVector
 begin
 
+lemma field_le_epsilon:
+  fixes x y :: "'a:: {number_ring,division_by_zero,ordered_field}"
+  assumes e: "(!!e. 0 < e ==> x \<le> y + e)"
+  shows "x \<le> y"
+proof (rule ccontr)
+  assume xy: "\<not> x \<le> y"
+  hence "(x-y)/2 > 0"
+    by (metis half_gt_zero le_iff_diff_le_0 linorder_not_le local.xy)
+  hence "x \<le> y + (x - y) / 2"
+    by (rule e [of "(x-y)/2"])
+  also have "... = (x - y + 2*y)/2"
+    by auto
+       (metis add_less_cancel_left add_numeral_0_right class_semiring.add_c xy e
+           diff_add_cancel gt_half_sum less_half_sum linorder_not_le number_of_Pls)
+  also have "... = (x + y) / 2" 
+    by auto
+  also have "... < x" using xy 
+    by auto
+  finally have "x<x" .
+  thus False
+    by auto 
+qed
+
+
 end