diff --git a/singular_free_tangent_sheaf_v2.txt b/singular_free_tangent_sheaf_v2.txt
index 84f0577..9fa3c43 100644
--- a/singular_free_tangent_sheaf_v2.txt
+++ b/singular_free_tangent_sheaf_v2.txt
@@ -174,7 +174,7 @@ isotropic precisely when q is a ramification point of F|_Gamma. On
 every two-dimensional quadratic space the polar form is x . y =
 (Q(x+y) - Q(x) - Q(y))/2, and x^2 abbreviates Q(x).
 
-A lemma produces six vectors in H^0(Gamma, Phi)^* with
+The following lemma produces six vectors in H^0(Gamma, Phi)^* with
 prescribed sum and normalization. If w = sum_j w_j and w_j^2 + w_j .
 w = 1 at each marked point, then
 
@@ -281,7 +281,7 @@ section s of Gamma + 4F on Sigma maps to
 with the normal differential first. This fixes the scalar in the
 canonical-root isomorphism for subsequent residue computations.
 
-A lemma computes the section spaces that transfer from Sigma to
+The following lemma computes the section spaces that transfer from Sigma to
 C and to Gamma. It uses the comparison theorem of Serre (Ann. Inst.
 Fourier 6, 1956, 1-42): algebraic and analytic section spaces of
 coherent sheaves on projective complex curves agree.
@@ -1318,7 +1318,7 @@ nonvanishing, and properness allows a further shrinking of X whose
 inverse image lies in this neighborhood. So v_1, v_2 frame
 T_{Y minus C}. []
 
-Counterexample. Proposition. There is a normal complex surface germ
+Proposition (counterexample). There is a normal complex surface germ
 (X, x) with an isolated singular point and free analytic derivation
 sheaf of rank two: