diff --git a/singular_free_tangent_sheaf_v2.txt b/singular_free_tangent_sheaf_v2.txt
index 6e9a8ca..a9ca25a 100644
--- a/singular_free_tangent_sheaf_v2.txt
+++ b/singular_free_tangent_sheaf_v2.txt
@@ -1493,11 +1493,11 @@ for r >= alpha(u). Choose alpha, beta, d >= 0 so that
 
   im(psi) intersect m_R^r R^t  is contained in  psi(m_R^{r - alpha} R^s),
   K intersect m_R^r R^s  is contained in  m_R^{r - beta} K,
-  I^{oplus s} intersect n^r P^s  is contained in
+  I^{⊕ s} intersect n^r P^s  is contained in
      { fC : C in n^{r - d} P^{s x t} },
 
 for all relevant r. In the last line, Artin-Rees applies to the map
-C |-> fC: P^{s x t} -> P^s with image I^{oplus s}; the relevant
+C |-> fC: P^{s x t} -> P^s with image I^{⊕ s}; the relevant
 filtration is the one induced from P^s.
 
 Choose l >= 0 with m_R^l K contained in J, and set
@@ -1528,7 +1528,7 @@ epsilon-bar to epsilon in n^{n+1} P^s. Then
 
   w = h - sum_j a_j d_j f - epsilon
 
-lies in I^{oplus s} intersect n^{n-c_0} P^s. The order of this
+lies in I^{⊕ s} intersect n^{n-c_0} P^s. The order of this
 ideal-valued remainder need only be n - c_0, since a derivative of a
 defining equation can be a unit. Artin-Rees on the ideal gives
 w = fC with C in n^{n-c} P^{s x t}. So