+ /3 width=5 by lfdeq_trans, tdeq_lfdeq_conf, fqup_strap1, tdeq_trans/
+]
+qed-.
+
+lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
+[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fqup, ex3_2_intro/
+| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
+ elim (tdeq_fqu_trans … H … HU21) -U1 #L0 #T #H1 #HTU #HL0
+ lapply (tdeq_lfdeq_div … HTU … HL0) -HL0 #HL0
+ elim (IH … HTU) -U #K2 #U1 #H2 #HUT1 #HKL2
+ elim (lfdeq_fqup_trans … H2 … HL0) -L #K #U #H2 #HU1 #HK2
+ lapply (tdeq_lfdeq_conf … HUT1 … HK2) -HK2 #HK2
+ /3 width=7 by lfdeq_trans, fqup_strap2, tdeq_trans, ex3_2_intro/
+]
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
+[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/