lemma feqg_cpx_trans_cpx (S):
reflexive … S → symmetric … S →
- â\88\80G1,G2,L1,L2,T1,T. â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG2,L2,Tâ\9d« →
- â\88\80T2. â\9dªG2,L2â\9d« â\8a¢ T â¬\88 T2 â\86\92 â\9dªG1,L1â\9d« ⊢ T1 ⬈ T2.
+ â\88\80G1,G2,L1,L2,T1,T. â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G2,L2,Tâ\9d© →
+ â\88\80T2. â\9d¨G2,L2â\9d© â\8a¢ T â¬\88 T2 â\86\92 â\9d¨G1,L1â\9d© ⊢ T1 ⬈ T2.
#S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (feqg_inv_gen_dx … H) -H // #H #HL12 #HT1 destruct
@(cpx_teqg_repl_reqg … HT2)
lemma feqg_cpx_trans_feqg (S):
reflexive … S → symmetric … S →
- â\88\80G1,G2,L1,L2,T1,T. â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG2,L2,Tâ\9d« →
- â\88\80T2. â\9dªG2,L2â\9d« â\8a¢ T â¬\88 T2 â\86\92 â\9dªG1,L1,T2â\9d« â\89\9b[S] â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T. â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G2,L2,Tâ\9d© →
+ â\88\80T2. â\9d¨G2,L2â\9d© â\8a¢ T â¬\88 T2 â\86\92 â\9d¨G1,L1,T2â\9d© â\89\9b[S] â\9d¨G2,L2,T2â\9d©.
#S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
elim (feqg_inv_gen_dx … H) -H // #H #HL12 #_ destruct
lapply (cpx_reqg_conf_dx … HT2 … HL12) -HT2 -HL12 // #HL12