Rename sage variables for better documentation

main.tex

@@ -165,12 +165,12 @@ $\chern(F)$, and so we can write:
 	c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
 
 	var("q", domain="real")
-	c_val = c_lower_bound + q
+	c_in_terms_of_q = c_lower_bound + q
 \end{sagesilent}
 
 \begin{equation}
 	\label{eqn-cintermsofm}
-	c=\chern_1(E) = \sage{c_val}
+	c=\chern_1(E) = \sage{c_in_terms_of_q}
 	\qquad 0 \leq q \leq \chern_1^{\beta_{-}}(F)
 \end{equation}
 
@@ -185,11 +185,11 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
 This condition expressed in terms of $R,C,D,r,c,d$ looks as follows:
 
 \begin{sagesilent}
-	positive_condition = - u.Q_tilt() - (v-u).Q_tilt() + v.Q_tilt()
+	bgmlv1 = - u.Q_tilt() - (v-u).Q_tilt() + v.Q_tilt()
 \end{sagesilent}
 
 \begin{equation}
-	\sage{(0 <= positive_condition.expand() )}
+	\sage{0 <= bgmlv1.expand() }
 \end{equation}
 
 
@@ -198,11 +198,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
 we get the following:
 
 \begin{sagesilent}
-	positive_condition = positive_condition.expand().subs(c == c_val)
+	bgmlv1_with_q = bgmlv1.expand().subs(c == c_in_terms_of_q)
 \end{sagesilent}
 
 \begin{equation}
-	\sage{(0 <= positive_condition) + 2*R*d - 4*d*r}
+	\sage{0 <= bgmlv1_with_q}
 \end{equation}
 
 
@@ -210,14 +210,16 @@ we get the following:
 This can be rearranged to express a bound on $d$ as follows:
 
 \begin{sagesilent}
-	var("r_alt",domain="real")
-	nc = (positive_condition.subs(r == r_alt + R/2)/r_alt).expand()
-	nc = ((0 > -nc) + 4*d)/4 # rearrange for d
-	nc = nc.subs(r_alt == r - R/2).expand() # resubs r back in
+	var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
+	bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
+	bgmlv1_d_ineq = ((0 > -bgmlv1_with_q_reparam) + 4*d)/4 # Rearrange for d
+	bgmlv1_d_ineq = bgmlv1_d_ineq.subs(r_alt == r - R/2).expand() # Resubstitute r back in
+
+	bgmlv1_d_lowerbound = bgmlv1_d_ineq.rhs() # Keep hold of lower bound for d
 \end{sagesilent}
 
 \begin{dmath}
-	\sage{nc}
+	\sage{bgmlv1_d_ineq}
 \end{dmath}