rewording

new_system_beppe.tex

@@ -102,8 +102,11 @@ The idea is to translate terms above in to explicitly annotated
 notrmal form and prove that a term above is well typed if and only if
 its translation is well typed for a suitable annotation.
 
-In particular the typing rule for cases will be obtained by combining
-the union rule with the case rule and using two subsumption rules. Intuitively
+In particular the typing rule for cases of CDuce will be obtained by combining
+the union rule with the case rule and using two subsumption
+rules. Intuitively we have that Cduce's $\tcase {x\,{:}{=}\,e} t {e_1}{e_2}$ is
+just syntactic sugar for  $\letexp x {e}{\tcase {x} t {e_1}{e_2}}$ and
+  we have that:
 \begin{mathpar}
   \inferrule*{
     \inferrule*{
@@ -120,8 +123,9 @@ the union rule with the case rule and using two subsumption rules. Intuitively
   }
   {\Gamma\vdash \letexp x {e}{\tcase {x} t {e_1}{e_2}:t'}}
 \end{mathpar}
-Note: $t_\circ =(t\wedge t_\circ)\vee(\neg t\wedge t_\circ$)
+[Note: $t_\circ =(t\wedge t_\circ)\vee(\neg t\wedge t_\circ$)]
 
+So we  find exactly the typing rule for Cduce's typecase.
 
 \subsubsection{Normal form terms with explicit annotations}