+
+lemma fpbg_fqu_trans (h): ∀G1,G,G2,L1,L,L2,T1,T,T2.
+ ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+#h #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
+/4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/
+qed-.
+
+(* Note: this is used in the closure proof *)
+lemma fpbg_fpbs_trans: ∀h,G,G2,L,L2,T,T2. ❪G,L,T❫ ≥[h] ❪G2,L2,T2❫ →
+ ∀G1,L1,T1. ❪G1,L1,T1❫ >[h] ❪G,L,T❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+#h #G #G2 #L #L2 #T #T2 #H @(fpbs_ind_dx … H) -G -L -T /3 width=5 by fpbg_fpbq_trans/
+qed-.
+
+(* Basic_2A1: uses: fpbg_fleq_trans *)
+lemma fpbg_feqx_trans: ∀h,G1,G,L1,L,T1,T. ❪G1,L1,T1❫ >[h] ❪G,L,T❫ →
+ ∀G2,L2,T2. ❪G,L,T❫ ≛ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+/3 width=5 by fpbg_fpbq_trans, fpbq_feqx/ qed-.
+
+(* Properties with t-bound rt-transition for terms **************************)
+
+lemma cpm_tneqx_cpm_fpbg (h) (G) (L):
+ ∀n1,T1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ❪G,L❫ ⊢ T ➡[n2,h] T2 → ❪G,L,T1❫ >[h] ❪G,L,T2❫.
+/4 width=5 by fpbq_fpbs, cpm_fpbq, cpm_fpb, ex2_3_intro/ qed.