--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2A/multiple/lleq_lleq.ma".
+include "basic_2A/multiple/fleq.ma".
+
+(* LAZY EQUIVALENCE FOR CLOSURES *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem fleq_trans: ∀l. tri_transitive … (fleq l).
+#l #G1 #G #L1 #L #T1 #T * -G -L -T
+#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/3 width=3 by lleq_trans, fleq_intro/
+qed-.
+
+theorem fleq_canc_sn: ∀G,G1,G2,L,L1,L2,T,T1,T2,l.
+ ⦃G, L, T⦄ ≡[l] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≡[l] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ qed-.
+
+theorem fleq_canc_dx: ∀G1,G2,G,L1,L2,L,T1,T2,T,l.
+ ⦃G1, L1, T1⦄ ≡[l] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≡[l] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≡[l] ⦃G2, L2, T2⦄.
+/3 width=5 by fleq_trans, fleq_sym/ qed-.