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diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
index 838ef0fed..ffd44c5aa 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
@@ -45,7 +45,7 @@ qed.
 
 (* Basic_1: was: ldrop_gen_refl *)
 lemma ldrop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5/ qed.
+/2 width=5/ qed-.
 
 fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
                           L2 = ⋆.
@@ -59,7 +59,7 @@ qed.
 
 (* Basic_1: was: ldrop_gen_sort *)
 lemma ldrop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
-/2 width=5/ qed.
+/2 width=5/ qed-.
 
 fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
                        ∀K,I,V. L1 = K. 𝕓{I} V → 
@@ -76,7 +76,7 @@ qed.
 lemma ldrop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
                     (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
                     (0 < e ∧ ↓[0, e - 1] K ≡ L2).
-/2/ qed.
+/2/ qed-.
 
 (* Basic_1: was: ldrop_gen_ldrop *)
 lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
@@ -84,7 +84,7 @@ lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
 #e #K #I #V #L2 #H #He
 elim (ldrop_inv_O1 … H) -H * // #H destruct -e;
 elim (lt_refl_false … He)
-qed.
+qed-.
 
 fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
                           ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
@@ -105,7 +105,7 @@ lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0
                        ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
                                 ↑[d - 1, e] V2 ≡ V1 & 
                                 L2 = K2. 𝕓{I} V2.
-/2/ qed.
+/2/ qed-.
 
 fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
                           ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
@@ -125,7 +125,7 @@ qed.
 lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
                        ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
                                 L1 = K1. 𝕓{I} V1.
-/2/ qed.
+/2/ qed-.
 
 (* Basic properties *********************************************************)
 
@@ -135,7 +135,7 @@ lemma ldrop_refl: ∀L. ↓[0, 0] L ≡ L.
 qed.
 
 lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
-                    ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+                      ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
 qed.
 
@@ -181,7 +181,7 @@ lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
   | @ldrop_ldrop >(plus_minus_m_m e 1) /2/
   ]
 ]
-qed.
+qed-.
 
 lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
 #L1 #L2 #d #e #H elim H -H L1 L2 d e // normalize
@@ -189,7 +189,7 @@ lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
   >(tw_lift … HV21) -HV21 /2/
 ]
-qed. 
+qed-. 
 
 lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
                                ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
@@ -201,7 +201,7 @@ lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
   | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
   ]
 ]
-qed.
+qed-.
 
 lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
 #L1 elim L1 -L1
@@ -213,7 +213,7 @@ lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
     >minus_le_minus_minus_comm //
   ]
 ]
-qed.
+qed-.
 
 (* Basic_1: removed theorems 49:
             ldrop_skip_flat