(* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
lemma feqg_fpbs (S) (G1) (G2) (L1) (L2) (T1) (T2):
reflexive … S → symmetric … S →
- â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=5 by fpb_fpbs, feqg_fpb/ qed.
(* Basic_2A1: uses: fpbs_lleq_trans *)
lemma fpbs_feqg_trans (S) (G) (L) (T):
reflexive … S → symmetric … S →
- â\88\80G1,L1,T1. â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG,L,Tâ\9d« →
- â\88\80G2,L2,T2. â\9dªG,L,Tâ\9d« â\89\9b[S] â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,L1,T1. â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G,L,Tâ\9d© →
+ â\88\80G2,L2,T2. â\9d¨G,L,Tâ\9d© â\89\9b[S] â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=9 by fpbs_strap1, feqg_fpb/ qed-.
(* Basic_2A1: uses: lleq_fpbs_trans *)
lemma feqg_fpbs_trans (S) (G) (L) (T):
reflexive … S → symmetric … S →
- â\88\80G2,L2,T2. â\9dªG,L,Tâ\9d« â\89¥ â\9dªG2,L2,T2â\9d« →
- â\88\80G1,L1,T1. â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG,L,Tâ\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G2,L2,T2. â\9d¨G,L,Tâ\9d© â\89¥ â\9d¨G2,L2,T2â\9d© →
+ â\88\80G1,L1,T1. â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G,L,Tâ\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=5 by fpbs_strap2, feqg_fpb/ qed-.
lemma teqg_fpbs_trans (S) (T):
reflexive … S → symmetric … S →
∀T1. T1 ≛[S] T →
- â\88\80G1,G2,L1,L2,T2. â\9dªG1,L1,Tâ\9d« â\89¥ â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,G2,L1,L2,T2. â\9d¨G1,L1,Tâ\9d© â\89¥ â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=8 by feqg_fpbs_trans, teqg_feqg/ qed-.
lemma fpbs_teqg_trans (S) (T):
reflexive … S → symmetric … S →
- â\88\80G1,G2,L1,L2,T1. â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,Tâ\9d« →
- â\88\80T2. T â\89\9b[S] T2 â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,G2,L1,L2,T1. â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,Tâ\9d© →
+ â\88\80T2. T â\89\9b[S] T2 â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=8 by fpbs_feqg_trans, teqg_feqg/ qed-.