diff --git a/noncoordinate_polynomial_fibre_v2.txt b/noncoordinate_polynomial_fibre_v2.txt
index aff1ab7..74207c4 100644
--- a/noncoordinate_polynomial_fibre_v2.txt
+++ b/noncoordinate_polynomial_fibre_v2.txt
@@ -27,7 +27,7 @@ maketitle
 
 Abstract: 
 We give an explicit polynomial F in four complex variables whose zero fibre is a
-copy of CC^3 and whose gradient vanishes at (2,0,-tfrac12,tfrac12). A
+copy of CC^3 and whose gradient vanishes at (2,0,-1/2,1/2). A
 polynomial whose gradient vanishes somewhere cannot be the first component of a
 polynomial automorphism of CC^4. This answers the Abhyankar--Sathaye question
 negatively in ambient dimension four; adjoining variables gives counterexamples
@@ -126,7 +126,7 @@ The polynomial F of (eq:F) satisfies
 [
   R/(F)congCC^{[3]}
    and 
-  grad F(2,0,-tfrac12,tfrac12)=0.
+  grad F(2,0,-1/2,1/2)=0.
 ]
 Consequently F is not a coordinate of R.
 
@@ -179,7 +179,7 @@ Modulo the relation hW=U^2+y, a direct expansion of (eq:xyz) gives
 ]
 and the relation U^3+hV=x becomes hG-yU=x. Hence
 
-  Ccong B[U,G,W]/(hG-yU-x, hW-U^2-y, y^2W-s+hG^2-2UyG).
+  C~= B[U,G,W]/(hG-yU-x, hW-U^2-y, y^2W-s+hG^2-2UyG).
 
 Put A=B[U,G]/(hG-yU-x) and in A set a=U^2+y, b=s-hG^2+2UyG. The last
 two relations of (eq:Clinear) read hW=a and y^2W=b. They are
@@ -196,7 +196,7 @@ y^2a=hb one gets hW_0=(rh+qy^2)a=a and y^2W_0=(rh+qy^2)b=b. Moreover
   W-W_0=r(hW-a)+q(y^2W-b),
 ]
 while hW-a=h(W-W_0) and y^2W-b=y^2(W-W_0). So in A[W] the ideal
-(hW-a,,y^2W-b) equals (W-W_0), and evaluating at W_0 gives Ccong A.
+(hW-a,,y^2W-b) equals (W-W_0), and evaluating at W_0 gives C~= A.
 
 It remains to identify A with B[T]. The B-algebra maps Ato B[T] and
 B[T]to A given by
@@ -211,7 +211,7 @@ are inverse. Indeed, in A the relation hG-yU=x and alpha h+beta y=1 give
 ]
 so substituting U=hT-beta x and G=yT+alpha x into alpha U+beta G
 returns T, and substituting T=alpha U+beta G into the first two returns U
-and G. Thus Acong B[T], and Ccong A.
+and G. Thus A~= B[T], and C~= A.
 
 In the original letters, the parameter is T=alpha U+beta(V+UW); its inverse
 recovers U and G from T, then W from W_0, then V=G-UW.
@@ -260,7 +260,7 @@ s=S(h,u,v,w). Under this substitution the relation x^2+y^3-sh becomes
 identically zero by (eq:cusp), and the relation h-1-p(x,y,s) becomes the
 equation F=0 of (eq:F). Therefore
 
-  R/(F)~= Ccong B[T]congCC[X,Y,T].
+  R/(F)~= C~= B[T]congCC[X,Y,T].
 
 This proves the first assertion of Theorem~thm:main: X, Y and T are
 polynomial coordinates on the zero fibre. Appendix~app:coordinates records