diff --git a/nearby_lagrangian_counterexample_v2.txt b/nearby_lagrangian_counterexample_v2.txt
index d04d435..07ad7bd 100644
--- a/nearby_lagrangian_counterexample_v2.txt
+++ b/nearby_lagrangian_counterexample_v2.txt
@@ -1,6 +1,6 @@
 ==============================================================================
 A COUNTEREXAMPLE TO THE NEARBY LAGRANGIAN CONJECTURE
-A complete, self-contained proof written to be read from top to bottom
+A complete, self-contained proof
 ==============================================================================
 
 What this document is.
@@ -31,7 +31,7 @@ What this document is.
     We use only plain ASCII. R^n is real n-space. S^k is the unit sphere in
     R^(k+1). <v,w> is the dot product. |x| is the length of a vector. A
     "smooth" map is infinitely differentiable. "Closed" always means compact
-    without boundary. We flag intuition boxes like this one as needed.
+    without boundary.
 
     
 
@@ -140,7 +140,7 @@ SECTION 1. BACKGROUND: COTANGENT BUNDLES, LAGRANGIANS, ISOTOPY
         T*Q gives the Lagrangian. This is a finite-dimensional, purely
         real-analytic gadget: the whole symplectic obstruction becomes a
         statement about critical points of ordinary smooth functions. That
-        is why high school students can follow the strategy: the geometry
+        is the reduction: the geometry
         has been converted to calculus.
 
     (B) STABLE TUBE DATA. The behaviour of F far from the origin (its