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diff --git a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
index 6dc1547bd..5c6a686e9 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/nf2/pr3.ma
@@ -18,24 +18,19 @@ include "basic_1/nf2/defs.ma".
 
 include "basic_1/pr3/pr3.ma".
 
-theorem nf2_pr3_unfold:
+lemma nf2_pr3_unfold:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr3 c t1 t2) \to ((nf2 c 
 t1) \to (eq T t1 t2)))))
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr3 c t1 
-t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
-t0)))) in (let TMP_4 \def (\lambda (t: T).(\lambda (H0: (nf2 c t)).(let TMP_2 
-\def (pr0_refl t) in (let TMP_3 \def (pr2_free c t t TMP_2) in (H0 t 
-TMP_3))))) in (let TMP_12 \def (\lambda (t0: T).(\lambda (t3: T).(\lambda 
-(H0: (pr2 c t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda 
-(H2: (((nf2 c t0) \to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def 
-H3 in (let TMP_5 \def (\lambda (t: T).(nf2 c t)) in (let TMP_6 \def (H4 t0 
-H0) in (let H5 \def (eq_ind T t3 TMP_5 H3 t0 TMP_6) in (let TMP_7 \def 
-(\lambda (t: T).(pr2 c t t0)) in (let TMP_8 \def (H4 t0 H0) in (let H6 \def 
-(eq_ind T t3 TMP_7 H0 t0 TMP_8) in (let TMP_9 \def (\lambda (t: T).(eq T t 
-t4)) in (let TMP_10 \def (H2 H5) in (let TMP_11 \def (H4 t0 H0) in (eq_ind_r 
-T t0 TMP_9 TMP_10 t3 TMP_11)))))))))))))))))) in (pr3_ind c TMP_1 TMP_4 
-TMP_12 t1 t2 H))))))).
+t2)).(pr3_ind c (\lambda (t: T).(\lambda (t0: T).((nf2 c t) \to (eq T t 
+t0)))) (\lambda (t: T).(\lambda (H0: (nf2 c t)).(H0 t (pr2_free c t t 
+(pr0_refl t))))) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H0: (pr2 c t3 
+t0)).(\lambda (t4: T).(\lambda (_: (pr3 c t0 t4)).(\lambda (H2: (((nf2 c t0) 
+\to (eq T t0 t4)))).(\lambda (H3: (nf2 c t3)).(let H4 \def H3 in (let H5 \def 
+(eq_ind T t3 (\lambda (t: T).(nf2 c t)) H3 t0 (H4 t0 H0)) in (let H6 \def 
+(eq_ind T t3 (\lambda (t: T).(pr2 c t t0)) H0 t0 (H4 t0 H0)) in (eq_ind_r T 
+t0 (\lambda (t: T).(eq T t t4)) (H2 H5) t3 (H4 t0 H0)))))))))))) t1 t2 H)))).
 
 theorem nf2_pr3_confluence:
  \forall (c: C).(\forall (t1: T).((nf2 c t1) \to (\forall (t2: T).((nf2 c t2) 
@@ -43,16 +38,13 @@ theorem nf2_pr3_confluence:
 \def
  \lambda (c: C).(\lambda (t1: T).(\lambda (H: (nf2 c t1)).(\lambda (t2: 
 T).(\lambda (H0: (nf2 c t2)).(\lambda (t: T).(\lambda (H1: (pr3 c t 
-t1)).(\lambda (H2: (pr3 c t t2)).(let TMP_1 \def (\lambda (t0: T).(pr3 c t2 
-t0)) in (let TMP_2 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let TMP_3 \def 
-(eq T t1 t2) in (let TMP_9 \def (\lambda (x: T).(\lambda (H3: (pr3 c t2 
-x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 x H4 H) in 
-(let TMP_4 \def (\lambda (t0: T).(pr3 c t1 t0)) in (let H5 \def (eq_ind_r T x 
-TMP_4 H4 t1 H_y) in (let TMP_5 \def (\lambda (t0: T).(pr3 c t2 t0)) in (let 
-H6 \def (eq_ind_r T x TMP_5 H3 t1 H_y) in (let H_y0 \def (nf2_pr3_unfold c t2 
-t1 H6 H0) in (let TMP_6 \def (\lambda (t0: T).(pr3 c t0 t1)) in (let H7 \def 
-(eq_ind T t2 TMP_6 H6 t1 H_y0) in (let TMP_7 \def (\lambda (t0: T).(eq T t1 
-t0)) in (let TMP_8 \def (refl_equal T t1) in (eq_ind_r T t1 TMP_7 TMP_8 t2 
-H_y0)))))))))))))) in (let TMP_10 \def (pr3_confluence c t t2 H2 t1 H1) in 
-(ex2_ind T TMP_1 TMP_2 TMP_3 TMP_9 TMP_10))))))))))))).
+t1)).(\lambda (H2: (pr3 c t t2)).(ex2_ind T (\lambda (t0: T).(pr3 c t2 t0)) 
+(\lambda (t0: T).(pr3 c t1 t0)) (eq T t1 t2) (\lambda (x: T).(\lambda (H3: 
+(pr3 c t2 x)).(\lambda (H4: (pr3 c t1 x)).(let H_y \def (nf2_pr3_unfold c t1 
+x H4 H) in (let H5 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t1 t0)) H4 t1 
+H_y) in (let H6 \def (eq_ind_r T x (\lambda (t0: T).(pr3 c t2 t0)) H3 t1 H_y) 
+in (let H_y0 \def (nf2_pr3_unfold c t2 t1 H6 H0) in (let H7 \def (eq_ind T t2 
+(\lambda (t0: T).(pr3 c t0 t1)) H6 t1 H_y0) in (eq_ind_r T t1 (\lambda (t0: 
+T).(eq T t1 t0)) (refl_equal T t1) t2 H_y0))))))))) (pr3_confluence c t t2 H2 
+t1 H1))))))))).