+
+lemma fqup_cpms_fwd_fpbg (h):
+ ∀G1,G2,L1,L2,T1,T. ❨G1,L1,T1❩ ⬂+ ❨G2,L2,T❩ →
+ ∀n,T2. ❨G2,L2❩ ⊢ T ➡*[h,n] T2 → ❨G1,L1,T1❩ > ❨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 ➡[h,n1] T → (T1 ≅ T → ⊥) →
+ ∀n2,T2. ❨G,L❩⊢ T ➡*[h,n2] T2 → T1 ≅ T2 → ❨G,L,T1❩ > ❨G,L,T1❩.
+#h #G #L #T1 #n1 #T #H1T1 #H2T1 #n2 #T2 #H1T2 #H2T12
+/4 width=10 by fpbc_fpbs_fpbg, cpms_fwd_fpbs, fpbs_teqg_trans, cpm_fwd_fpbc, teqx_sym/
+qed-.