+
+(* the main properties ******************************************************)
+
+axiom thin_conf_ge: ∀d1,e1,L,L1. ↓[d1,e1] L ≡ L1 →
+ ∀e2,L2. ↓[0,e2] L ≡ L2 → d1 + e1 ≤ e2 → ↓[0,e2-e1] L1 ≡ L2.
+
+axiom thin_conf_lt: ∀d1,e1,L,L1. ↓[d1,e1] L ≡ L1 →
+ ∀e2,K2,I,V2. ↓[0,e2] L ≡ K2. 𝕓{I} V2 →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ↓[0,e2] L1 ≡ K1. 𝕓{I} V1 & ↓[d,e1] K2 ≡ K1 & ↑[d,e1] V1 ≡ V2.