lemma fpbg_fqu_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ > ❪G,L,T❫ → ❪G,L,T❫ ⬂ ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫. #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 /4 width=5 by fpbg_fpbq_trans, fpbq_fquq, fqu_fquq/ qed-. lemma fpbq_fpbg_trans: ∀G1,G,G2,L1,L,L2,T1,T,T2. ❪G1,L1,T1❫ ≻ ❪G,L,T❫ → ❪G,L,T❫ > ❪G2,L2,T2❫ → ❪G1,L1,T1❫ > ❪G2,L2,T2❫. /3 width=5 by fpbg_fwd_fpbs, ex2_3_intro/ qed-.