-(*
-theorem SUBSETS_full: ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
- intros; exists;
-
-*)
\ No newline at end of file
+theorem BP_to_OBP_full:
+ ∀S,T.∀f. exT22 ? (λg. map_arrows2 ?? BP_to_OBP S T g = f).
+ intros;
+ cases (POW_full (concr S) (concr T) (Oconcr_rel ?? f)) (gc Hgc);
+ cases (POW_full (form S) (form T) (Oform_rel ?? f)) (gf Hgf);
+ exists[
+ constructor 1; [apply gc|apply gf]
+ apply (POW_faithful);
+ apply (let xxxx ≝POW in .= respects_comp2 ?? POW (concr S) (concr T) (form T) gc (rel T));
+ apply rule (.= Hgc‡#);
+ apply (.= Ocommute ?? f);
+ apply (.= #‡Hgf^-1);
+ apply (let xxxx ≝POW in (respects_comp2 ?? POW (concr S) (form S) (form T) (rel S) gf)^-1)]
+ split;
+ [ change in match or_f_minus_star_ with (λq,w,x.fun12 ?? (or_f_minus_star q w) x);
+ | change in match or_f_minus_ with (λq,w,x.fun12 ?? (or_f_minus q w) x);
+ | change in match or_f_ with (λq,w,x.fun12 ?? (or_f q w) x);
+ | change in match or_f_star_ with (λq,w,x.fun12 ?? (or_f_star q w) x);]
+ simplify; apply (†(Hgc‡#));
+qed.