diff --git a/artin_parabolic_intersections_v2.txt b/artin_parabolic_intersections_v2.txt
index 881fcf0..1de3b45 100644
--- a/artin_parabolic_intersections_v2.txt
+++ b/artin_parabolic_intersections_v2.txt
@@ -681,7 +681,7 @@ to f).  So D = X and z = 1, giving
 
 PROOF OF THE MAIN THEOREM.
 
-  STEP 1.  Set x = h^{-1} g in A (so g = h x^{-1}).  Then
+  STEP 1.  Set x = h^{-1} g in A (so g = h x).  Then
 
     g A_X g^{-1}  ∩  h A_Y h^{-1}
       = h ( x A_X x^{-1}  ∩  A_Y ) h^{-1}.
@@ -954,21 +954,11 @@ EXAMPLE 5:  a nonspherical case with braid relations
   Take P = <a, b> and Q = <b, c>.  Both are Braid_3-type.  They
   share <b>.  Is the intersection bigger than <b>?
 
-  Suppose x ∈ <a, b> ∩ <b, c> is not in <b>.  Then x is a
-  nonempty reduced word in {a, b, a^{-1}, b^{-1}} (in the first
-  subgroup) that is also a reduced word in {b, c, b^{-1}, c^{-1}}
-  (in the second).  In the free product structure underlying A, an
-  element is uniquely determined by its normal form.  An element
-  in the subgroup generated by a, b has normal form using only
-  a-appearances and b-appearances; an element in the subgroup
-  generated by b, c has normal form using only b-appearances and
-  c-appearances.  An element in BOTH must use only b-appearances
-  (else it would have an a or c appearance, contradicting
-  membership in the other subgroup; the subgroup <a, b> really
-  contains elements with an "a" in the normal form, and similarly
-  for c).  So x ∈ <b>.  Therefore:
-
-    P ∩ Q = <b>.
+  By van der Lek's 1983 theorem (for standard parabolics defined
+  by subdiagrams), A_{X} ∩ A_{Y} = A_{X ∩ Y} for any X, Y ⊆ S.
+  Here X = {a, b}, Y = {b, c}, X ∩ Y = {b}.  So:
+
+    P ∩ Q = A_{a,b} ∩ A_{b,c} = A_b = <b>.
 
   Parabolic (rank 1).                                    GOOD