+
+lemma fqup_cpms_fwd_fpbg (h):
+ ∀G1,G2,L1,L2,T1,T. ❪G1,L1,T1❫ ⬂+ ❪G2,L2,T❫ →
+ ∀n,T2. ❪G2,L2❫ ⊢ T ➡*[n,h] T2 → ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫.
+/3 width=5 by cpms_fwd_fpbs, fqup_fpbg, fpbg_fpbs_trans/ qed-.
+
+lemma cpm_tneqx_cpm_cpms_teqx_sym_fwd_fpbg (h) (G) (L) (T1):
+ ∀n1,T. ❪G,L❫ ⊢ T1 ➡[n1,h] T → (T1 ≛ T → ⊥) →
+ ∀n2,T2. ❪G,L❫⊢ T ➡*[n2,h] T2 → T1 ≛ T2 → ❪G,L,T1❫ >[h] ❪G,L,T1❫.
+#h #G #L #T1 #n1 #T #H1T1 #H2T1 #n2 #T2 #H1T2 #H2T12
+/4 width=7 by cpms_fwd_fpbs, cpm_fpb, fpbs_teqx_trans, teqx_sym, ex2_3_intro/
+qed-.