elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/
| #i1 #i #H <(tdeq_inv_lref1 … H) -H //
| #l1 #l #H <(tdeq_inv_gref1 … H) -H //
-| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H destruct
+| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/
]
qed-.
theorem tdeq_repl: ∀h,o,T1,T2. T1 ≡[h, o] T2 →
∀U1. T1 ≡[h, o] U1 → ∀U2. T2 ≡[h, o] U2 → U1 ≡[h, o] U2.
/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
+
+(* Negated main properies ***************************************************)
+
+theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≡[h, o] T → ∀T2. (T ≡[h, o] T2 → ⊥) →
+ T1 ≡[h, o] T2 → ⊥.
+/3 width=3 by tdeq_canc_sn/ qed-.
+
+theorem tndeq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≡[h, o] T → ⊥) → ∀T2. T2 ≡[h, o] T →
+ T1 ≡[h, o] T2 → ⊥.
+/3 width=3 by tdeq_trans/ qed-.