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diff --git a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
index 05744e7d0..e00736172 100644
--- a/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_1/sn3/nf2.ma
@@ -20,55 +20,41 @@ include "basic_1/nf2/dec.ma".
 
 include "basic_1/nf2/pr3.ma".
 
-theorem sn3_nf2:
+lemma sn3_nf2:
  \forall (c: C).(\forall (t: T).((nf2 c t) \to (sn3 c t)))
 \def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(let TMP_7 \def 
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (nf2 c t)).(sn3_sing c t 
 (\lambda (t2: T).(\lambda (H0: (((eq T t t2) \to (\forall (P: 
 Prop).P)))).(\lambda (H1: (pr3 c t t2)).(let H_y \def (nf2_pr3_unfold c t t2 
-H1 H) in (let TMP_1 \def (\lambda (t0: T).(pr3 c t t0)) in (let H2 \def 
-(eq_ind_r T t2 TMP_1 H1 t H_y) in (let TMP_2 \def (\lambda (t0: T).((eq T t 
-t0) \to (\forall (P: Prop).P))) in (let H3 \def (eq_ind_r T t2 TMP_2 H0 t 
-H_y) in (let TMP_3 \def (\lambda (t0: T).(sn3 c t0)) in (let TMP_4 \def 
-(refl_equal T t) in (let TMP_5 \def (sn3 c t) in (let TMP_6 \def (H3 TMP_4 
-TMP_5) in (eq_ind T t TMP_3 TMP_6 t2 H_y))))))))))))) in (sn3_sing c t 
-TMP_7)))).
+H1 H) in (let H2 \def (eq_ind_r T t2 (\lambda (t0: T).(pr3 c t t0)) H1 t H_y) 
+in (let H3 \def (eq_ind_r T t2 (\lambda (t0: T).((eq T t t0) \to (\forall (P: 
+Prop).P))) H0 t H_y) in (eq_ind T t (\lambda (t0: T).(sn3 c t0)) (H3 
+(refl_equal T t) (sn3 c t)) t2 H_y)))))))))).
 
-theorem nf2_sn3:
+lemma nf2_sn3:
  \forall (c: C).(\forall (t: T).((sn3 c t) \to (ex2 T (\lambda (u: T).(pr3 c 
 t u)) (\lambda (u: T).(nf2 c u)))))
 \def
- \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(let TMP_3 \def 
-(\lambda (t0: T).(let TMP_1 \def (\lambda (u: T).(pr3 c t0 u)) in (let TMP_2 
-\def (\lambda (u: T).(nf2 c u)) in (ex2 T TMP_1 TMP_2)))) in (let TMP_32 \def 
+ \lambda (c: C).(\lambda (t: T).(\lambda (H: (sn3 c t)).(sn3_ind c (\lambda 
+(t0: T).(ex2 T (\lambda (u: T).(pr3 c t0 u)) (\lambda (u: T).(nf2 c u)))) 
 (\lambda (t1: T).(\lambda (_: ((\forall (t2: T).((((eq T t1 t2) \to (\forall 
 (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2)))))).(\lambda (H1: ((\forall 
 (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to 
 (ex2 T (\lambda (u: T).(pr3 c t2 u)) (\lambda (u: T).(nf2 c u)))))))).(let 
-H_x \def (nf2_dec c t1) in (let H2 \def H_x in (let TMP_4 \def (nf2 c t1) in 
-(let TMP_5 \def (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in 
-(let TMP_6 \def (\lambda (t2: T).(pr2 c t1 t2)) in (let TMP_7 \def (ex2 T 
-TMP_5 TMP_6) in (let TMP_8 \def (\lambda (u: T).(pr3 c t1 u)) in (let TMP_9 
-\def (\lambda (u: T).(nf2 c u)) in (let TMP_10 \def (ex2 T TMP_8 TMP_9) in 
-(let TMP_14 \def (\lambda (H3: (nf2 c t1)).(let TMP_11 \def (\lambda (u: 
-T).(pr3 c t1 u)) in (let TMP_12 \def (\lambda (u: T).(nf2 c u)) in (let 
-TMP_13 \def (pr3_refl c t1) in (ex_intro2 T TMP_11 TMP_12 t1 TMP_13 H3))))) 
-in (let TMP_31 \def (\lambda (H3: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to 
-(\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))).(let TMP_15 \def 
-(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) in (let TMP_16 
-\def (\lambda (t2: T).(pr2 c t1 t2)) in (let TMP_17 \def (\lambda (u: T).(pr3 
-c t1 u)) in (let TMP_18 \def (\lambda (u: T).(nf2 c u)) in (let TMP_19 \def 
-(ex2 T TMP_17 TMP_18) in (let TMP_30 \def (\lambda (x: T).(\lambda (H4: (((eq 
-T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y 
-\def (H1 x H4) in (let TMP_20 \def (pr3_pr2 c t1 x H5) in (let H6 \def (H_y 
-TMP_20) in (let TMP_21 \def (\lambda (u: T).(pr3 c x u)) in (let TMP_22 \def 
-(\lambda (u: T).(nf2 c u)) in (let TMP_23 \def (\lambda (u: T).(pr3 c t1 u)) 
-in (let TMP_24 \def (\lambda (u: T).(nf2 c u)) in (let TMP_25 \def (ex2 T 
-TMP_23 TMP_24) in (let TMP_29 \def (\lambda (x0: T).(\lambda (H7: (pr3 c x 
-x0)).(\lambda (H8: (nf2 c x0)).(let TMP_26 \def (\lambda (u: T).(pr3 c t1 u)) 
-in (let TMP_27 \def (\lambda (u: T).(nf2 c u)) in (let TMP_28 \def (pr3_sing 
-c x t1 H5 x0 H7) in (ex_intro2 T TMP_26 TMP_27 x0 TMP_28 H8))))))) in 
-(ex2_ind T TMP_21 TMP_22 TMP_25 TMP_29 H6))))))))))))) in (ex2_ind T TMP_15 
-TMP_16 TMP_19 TMP_30 H3)))))))) in (or_ind TMP_4 TMP_7 TMP_10 TMP_14 TMP_31 
-H2))))))))))))))) in (sn3_ind c TMP_3 TMP_32 t H))))).
+H_x \def (nf2_dec c t1) in (let H2 \def H_x in (or_ind (nf2 c t1) (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2))) (ex2 T (\lambda (u: T).(pr3 c t1 u)) (\lambda (u: T).(nf2 
+c u))) (\lambda (H3: (nf2 c t1)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) t1 (pr3_refl c t1) H3)) (\lambda (H3: (ex2 T 
+(\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: 
+T).(pr2 c t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall 
+(P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)) (ex2 T (\lambda (u: T).(pr3 c 
+t1 u)) (\lambda (u: T).(nf2 c u))) (\lambda (x: T).(\lambda (H4: (((eq T t1 
+x) \to (\forall (P: Prop).P)))).(\lambda (H5: (pr2 c t1 x)).(let H_y \def (H1 
+x H4) in (let H6 \def (H_y (pr3_pr2 c t1 x H5)) in (ex2_ind T (\lambda (u: 
+T).(pr3 c x u)) (\lambda (u: T).(nf2 c u)) (ex2 T (\lambda (u: T).(pr3 c t1 
+u)) (\lambda (u: T).(nf2 c u))) (\lambda (x0: T).(\lambda (H7: (pr3 c x 
+x0)).(\lambda (H8: (nf2 c x0)).(ex_intro2 T (\lambda (u: T).(pr3 c t1 u)) 
+(\lambda (u: T).(nf2 c u)) x0 (pr3_sing c x t1 H5 x0 H7) H8)))) H6)))))) H3)) 
+H2)))))) t H))).