diff --git a/main.tex b/main.tex
index 03dd203d213224bfd100d7aae1d35bf99736ac7c..62d6656c6f21edb3d2a742f4a9bdd7af21c064d4 100644
--- a/main.tex
+++ b/main.tex
@@ -462,16 +462,20 @@ bgmlv3_d_upperbound_exp_term = (
 # Verify the simplified forms of the terms that will be mentioned in text
 
 assert bgmlv3_d_upperbound_const_term == ( 
+  bgmlv3_d_upperbound_const_term_alt1 :=
 	v.twist(beta_min).ch[2]
 	+ beta_min*q
 ).expand()
 
 assert bgmlv3_d_upperbound_exp_term == (
+  bgmlv3_d_upperbound_const_term_alt2 :=
+	(
 		R*v.twist(beta_min).ch[2]
 		+ (C - q)^2/2
 		+ R*beta_min*q
 		- D*R
 	)/(r-R)
+)
 \end{sagesilent}