ITP001 Axioms: ITP033^7.ax
%------------------------------------------------------------------------------
% File : ITP033^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : set_relation.ax [Gau19]
% : HL4033^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 274 ( 58 unt; 96 typ; 0 def)
% Number of atoms : 1072 ( 122 equ; 24 cnn)
% Maximal formula atoms : 26 ( 3 avg)
% Number of connectives : 4808 ( 24 ~; 9 |; 163 &;4441 @)
% ( 34 <=>; 137 =>; 0 <=; 0 <~>)
% Maximal formula depth : 41 ( 10 avg;4441 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 779 ( 779 >; 0 *; 0 +; 0 <<)
% Number of symbols : 96 ( 94 usr; 3 con; 0-7 aty)
% Number of variables : 903 ( 64 ^ 706 !; 28 ?; 903 :)
% ( 105 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eoption_2Eoption,type,
tyop_2Eoption_2Eoption: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epred__set_2ECHOICE,type,
c_2Epred__set_2ECHOICE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Epair_2ECURRY,type,
c_2Epair_2ECURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) > A_27a > A_27b > A_27c ) ).
thf(c_2Epred__set_2EDELETE,type,
c_2Epred__set_2EDELETE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDISJOINT,type,
c_2Epred__set_2EDISJOINT:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2EFST,type,
c_2Epair_2EFST:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINJ,type,
c_2Epred__set_2EINJ:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2ELET,type,
c_2Ebool_2ELET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > A_27a > A_27b ) ).
thf(c_2Eoption_2ENONE,type,
c_2Eoption_2ENONE:
!>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Erelation_2EO,type,
c_2Erelation_2EO:
!>[A_27g: $tType,A_27h: $tType,A_27k: $tType] : ( ( A_27h > A_27k > $o ) > ( A_27g > A_27h > $o ) > A_27g > A_27k > $o ) ).
thf(c_2Erelation_2ERDOM,type,
c_2Erelation_2ERDOM:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EREL__RESTRICT,type,
c_2Epred__set_2EREL__RESTRICT:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ERRANGE,type,
c_2Erelation_2ERRANGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > A_27b > $o ) ).
thf(c_2Eset__relation_2ERREFL__EXP,type,
c_2Eset__relation_2ERREFL__EXP:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Eset__relation_2ERRUNIV,type,
c_2Eset__relation_2ERRUNIV:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Erelation_2ERSUBSET,type,
c_2Erelation_2ERSUBSET:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > $o ) ).
thf(c_2Erelation_2ERUNION,type,
c_2Erelation_2ERUNION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( A_27a > A_27b > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Epred__set_2ESING,type,
c_2Epred__set_2ESING:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2ESND,type,
c_2Epair_2ESND:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27b ) ).
thf(c_2Eoption_2ESOME,type,
c_2Eoption_2ESOME:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2ESWAP,type,
c_2Epair_2ESWAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2ETC,type,
c_2Erelation_2ETC:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $o ) ).
thf(c_2Eoption_2ETHE,type,
c_2Eoption_2ETHE:
!>[A_27a: $tType] : ( ( tyop_2Eoption_2Eoption @ A_27a ) > A_27a ) ).
thf(c_2Epair_2EUNCURRY,type,
c_2Epair_2EUNCURRY:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Erelation_2EWF,type,
c_2Erelation_2EWF:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWeakLinearOrder,type,
c_2Erelation_2EWeakLinearOrder:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Erelation_2EWeakOrder,type,
c_2Erelation_2EWeakOrder:
!>[A_27g: $tType] : ( ( A_27g > A_27g > $o ) > $o ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Eset__relation_2Eacyclic,type,
c_2Eset__relation_2Eacyclic:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o ) ).
thf(c_2Eset__relation_2Eall__choices,type,
c_2Eset__relation_2Eall__choices:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Eantisym,type,
c_2Eset__relation_2Eantisym:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o ) ).
thf(c_2Erelation_2Eantisymmetric,type,
c_2Erelation_2Eantisymmetric:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Echain,type,
c_2Eset__relation_2Echain:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o ) ).
thf(c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Edomain,type,
c_2Eset__relation_2Edomain:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > A_27a > $o ) ).
thf(c_2Eset__relation_2Efchains,type,
c_2Eset__relation_2Efchains:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Efinite__prefixes,type,
c_2Eset__relation_2Efinite__prefixes:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Eset__relation_2Eget__min,type,
c_2Eset__relation_2Eget__min:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ) > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Eirreflexive,type,
c_2Eset__relation_2Eirreflexive:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Elinear__order,type,
c_2Eset__relation_2Elinear__order:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Emaximal__elements,type,
c_2Eset__relation_2Emaximal__elements:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Eset__relation_2Eminimal__elements,type,
c_2Eset__relation_2Eminimal__elements:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Eset__relation_2Enth__min,type,
c_2Eset__relation_2Enth__min:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ) > tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).
thf(c_2Eset__relation_2Enum__order,type,
c_2Eset__relation_2Enum__order:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27a > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Epartial__order,type,
c_2Eset__relation_2Epartial__order:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Eper,type,
c_2Eset__relation_2Eper:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Eset__relation_2Eper__restrict,type,
c_2Eset__relation_2Eper__restrict:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Erange,type,
c_2Eset__relation_2Erange:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Eset__relation_2Ercomp,type,
c_2Eset__relation_2Ercomp:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > $o ) > ( ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Erelation_2Ereflexive,type,
c_2Erelation_2Ereflexive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Ereflexive,type,
c_2Eset__relation_2Ereflexive:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Erel__to__reln,type,
c_2Eset__relation_2Erel__to__reln:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) ).
thf(c_2Eset__relation_2Ereln__to__rel,type,
c_2Eset__relation_2Ereln__to__rel:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o ) > A_27a > A_27b > $o ) ).
thf(c_2Eset__relation_2Errestrict,type,
c_2Eset__relation_2Errestrict:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Estrict,type,
c_2Eset__relation_2Estrict:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Estrict__linear__order,type,
c_2Eset__relation_2Estrict__linear__order:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Etc,type,
c_2Eset__relation_2Etc:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive:
!>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > $o ) ).
thf(c_2Eset__relation_2Etransitive,type,
c_2Eset__relation_2Etransitive:
!>[A_27a: $tType] : ( ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o ) ).
thf(c_2Eset__relation_2Euniv__reln,type,
c_2Eset__relation_2Euniv__reln:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ).
thf(c_2Eset__relation_2Eupper__bounds,type,
c_2Eset__relation_2Eupper__bounds:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > $o ) > ( ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o ) > A_27a > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Eset__relation_2Edomain__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V0r )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V1x
@ ( c_2Ebool_2E_3F @ A_27b
@ ^ [V2y: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Erange__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o] :
( ( c_2Eset__relation_2Erange @ A_27a @ A_27b @ V0r )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V1y: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V1y
@ ( c_2Ebool_2E_3F @ A_27b
@ ^ [V2x: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2x @ V1y ) @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Errestrict__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V1s )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V2x: A_27a,V3y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Ercomp__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) > $o] :
( ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c @ V0r1 @ V1r2 )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V2x: A_27a,V3y: A_27b] :
( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y )
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V4z: A_27c] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ ( c_2Epair_2E_2C @ A_27a @ A_27c @ V2x @ V4z ) @ V0r1 ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) @ ( c_2Epair_2E_2C @ A_27c @ A_27b @ V4z @ V3y ) @ V1r2 ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Estrict__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Estrict @ A_27a @ V0r )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V1x: A_27a,V2y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ A_27a @ V1x @ V2y ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Euniv__reln__def,axiom,
! [A_27a: $tType,V0xs: A_27a > $o] :
( ( c_2Eset__relation_2Euniv__reln @ A_27a @ V0xs )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V1x1: A_27a,V2x2: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x1 @ V2x2 ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V1x1 @ V0xs ) @ ( c_2Ebool_2EIN @ A_27a @ V2x2 @ V0xs ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1s: A_27b > $o] :
( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V1s )
<=> ! [V2e: A_27b] :
( ( c_2Ebool_2EIN @ A_27b @ V2e @ V1s )
=> ( c_2Epred__set_2EFINITE @ A_27a
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V3e_27: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V3e_27 @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V3e_27 @ V2e ) @ V0r ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__def,axiom,
! [A_27a: $tType] :
( ( c_2Eset__relation_2Etc @ A_27a )
= ( ^ [V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1a0: tyop_2Epair_2Eprod @ A_27a @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o )
@ ^ [V2tc_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V3a0: tyop_2Epair_2Eprod @ A_27a @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4x: A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V5y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V3a0 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V5y ) ) @ ( V0r @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V5y ) ) ) ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V6x: A_27a] :
( c_2Ebool_2E_3F @ A_27a
@ ^ [V7y: A_27a] :
( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V3a0 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6x @ V7y ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V8z: A_27a] : ( c_2Ebool_2E_2F_5C @ ( V2tc_27 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6x @ V8z ) ) @ ( V2tc_27 @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V8z @ V7y ) ) ) ) ) ) ) )
@ ( V2tc_27 @ V3a0 ) ) )
@ ( V2tc_27 @ V1a0 ) ) ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r )
<=> ! [V1x: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V1x ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Ereflexive__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Ereflexive @ A_27a @ V0r @ V1s )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V2x ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eirreflexive__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Eirreflexive @ A_27a @ V0r @ V1s )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V2x ) @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Etransitive__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
<=> ! [V1x: A_27a,V2y: A_27a,V3z: A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2y @ V3z ) @ V0r ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3z ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eantisym__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eantisym @ A_27a @ V0r )
<=> ! [V1x: A_27a,V2y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2y @ V1x ) @ V0r ) )
=> ( V1x = V2y ) ) ) ).
thf(thm_2Eset__relation_2Epartial__order__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
& ( c_2Eset__relation_2Ereflexive @ A_27a @ V0r @ V1s )
& ( c_2Eset__relation_2Eantisym @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
& ( c_2Eset__relation_2Eantisym @ A_27a @ V0r )
& ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) )
=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
| ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V2x ) @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) @ V1s )
& ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
& ! [V2x: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V2x ) @ V0r ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ V1s )
& ( (~) @ ( V3x = V4y ) ) )
=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V4y ) @ V0r )
| ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4y @ V3x ) @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r )
= ( ^ [V1x: A_27a,V2y: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V0R )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o )
@ ^ [V1x: A_27a,V2y: A_27b] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V2y ) @ ( V0R @ V1x @ V2y ) ) ) ) ) ).
thf(thm_2Eset__relation_2ERRUNIV__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eset__relation_2ERRUNIV @ A_27a @ V0s )
= ( ^ [V1x: A_27a,V2y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V1x @ V0s ) @ ( c_2Ebool_2EIN @ A_27a @ V2y @ V0s ) ) ) ) ).
thf(thm_2Eset__relation_2ERREFL__EXP__def,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ V0R @ V1s )
= ( c_2Erelation_2ERUNION @ A_27a @ A_27a @ V0R
@ ^ [V2x: A_27a,V3y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ A_27a @ V2x @ V3y ) @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) ) ) ).
thf(thm_2Eset__relation_2Emaximal__elements__def,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0xs @ V1r )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V2x
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0xs )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x_27: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x_27 @ V0xs ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3x_27 ) @ V1r ) ) @ ( c_2Emin_2E_3D @ A_27a @ V2x @ V3x_27 ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__elements__def,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ V1r )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V2x
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V0xs )
@ ( c_2Ebool_2E_21 @ A_27a
@ ^ [V3x_27: A_27a] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3x_27 @ V0xs ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x_27 @ V2x ) @ V1r ) ) @ ( c_2Emin_2E_3D @ A_27a @ V2x @ V3x_27 ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Echain__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Echain @ A_27a @ V0s @ V1r )
<=> ! [V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V0s ) )
=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V1r )
| ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V2x ) @ V1r ) ) ) ) ).
thf(thm_2Eset__relation_2Eupper__bounds__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: A_27b > $o,V1r: ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27b @ V0s @ V1r )
= ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V2x: A_27a] :
( c_2Epair_2E_2C @ A_27a @ $o @ V2x
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b @ V1r ) )
@ ( c_2Ebool_2E_21 @ A_27b
@ ^ [V3y: A_27b] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ A_27b @ V3y @ V0s ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V3y @ V2x ) @ V1r ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Efchains__def,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Efchains @ A_27a @ V0r )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1k: A_27a > $o] :
( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1k
@ ( c_2Ebool_2E_2F_5C @ ( c_2Eset__relation_2Echain @ A_27a @ V1k @ V0r )
@ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V1k @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
@ ( c_2Ebool_2E_21 @ ( A_27a > $o )
@ ^ [V2C: A_27a > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2E_2F_5C @ ( c_2Eset__relation_2Echain @ A_27a @ V2C @ V0r ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2ESUBSET @ A_27a @ V2C @ V1k ) @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V2C @ V0r ) @ V2C ) @ V1k ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) @ ( c_2Ebool_2EIN @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V2C @ V0r ) @ V2C ) ) @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ ( c_2Epred__set_2EINTER @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V2C @ V0r ) @ V2C ) @ V1k ) @ V0r ) ) ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Eper__def,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1xss: ( A_27a > $o ) > $o] :
( ( c_2Eset__relation_2Eper @ A_27a @ V0xs @ V1xss )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1xss ) @ V0xs )
& ( (~) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V1xss ) )
& ! [V2xs1: A_27a > $o,V3xs2: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2xs1 @ V1xss )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3xs2 @ V1xss )
& ( (~) @ ( V2xs1 = V3xs2 ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ V2xs1 @ V3xs2 ) ) ) ) ).
thf(thm_2Eset__relation_2Eper__restrict__def,axiom,
! [A_27a: $tType,V0xss: ( A_27a > $o ) > $o,V1xs: A_27a > $o] :
( ( c_2Eset__relation_2Eper__restrict @ A_27a @ V0xss @ V1xs )
= ( c_2Epred__set_2EDELETE @ ( A_27a > $o )
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2xs_27: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EINTER @ A_27a @ V2xs_27 @ V1xs ) @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2xs_27 @ V0xss ) ) )
@ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ).
thf(thm_2Eset__relation_2Eall__choices__def,axiom,
! [A_27a: $tType,V0xss: ( A_27a > $o ) > $o] :
( ( c_2Eset__relation_2Eall__choices @ A_27a @ V0xss )
= ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( ( A_27a > $o ) > A_27a )
@ ^ [V1choice: ( A_27a > $o ) > A_27a] :
( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ A_27a @ V1choice @ V0xss )
@ ( c_2Ebool_2E_21 @ ( A_27a > $o )
@ ^ [V2xs: A_27a > $o] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2xs @ V0xss ) @ ( c_2Ebool_2EIN @ A_27a @ ( V1choice @ V2xs ) @ V2xs ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Enum__order__def,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Enum__order @ A_27a @ V0f @ V1s )
= ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V2x: A_27a,V3y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) @ ( c_2Earithmetic_2E_3C_3D @ ( V0f @ V2x ) @ ( V0f @ V3y ) ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Eget__min__def,axiom,
! [A_27a: $tType,V0r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eget__min @ A_27a @ V0r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V1s @ V2r ) )
= ( c_2Ebool_2ELET @ ( A_27a > $o ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V3mins: A_27a > $o] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Epred__set_2ESING @ A_27a @ V3mins ) @ ( c_2Eoption_2ESOME @ A_27a @ ( c_2Epred__set_2ECHOICE @ A_27a @ V3mins ) ) @ ( c_2Eoption_2ENONE @ A_27a ) )
@ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V1s @ V2r ) @ V0r_27 ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__bigunion,axiom,
! [A_27a: $tType,V0rs: ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > $o] :
( ( ! [V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Ebool_2EIN @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V1r @ V0rs )
& ( c_2Ebool_2EIN @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V2r_27 @ V0rs )
& ( (~) @ ( V1r = V2r_27 ) ) )
=> ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V2r_27 ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V2r_27 ) ) ) )
& ! [V3r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Ebool_2EIN @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V3r @ V0rs )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ V3r ) ) )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Epred__set_2EBIGUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0rs ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__SWAP,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) @ V0r ) )
= ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r ) ) ).
thf(thm_2Eset__relation_2Eacyclic__irreflexive,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r )
=> ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V1x ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__rrestrict,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V1s ) ) ) ).
thf(thm_2Eset__relation_2Etc__mono,axiom,
! [A_27a: $tType,V0s: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V1r @ V0s )
=> ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Etc__idemp,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
= ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ).
thf(thm_2Eset__relation_2Esubset__tc,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ).
thf(thm_2Eset__relation_2Etc__closure,axiom,
! [A_27a: $tType,V0s: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V1r @ ( c_2Eset__relation_2Etc @ A_27a @ V0s ) )
=> ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Etc__ind,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( V1tc_27 @ V4x @ V6z )
& ( V1tc_27 @ V6z @ V5y ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Erextension,axiom,
! [A_27a: $tType,A_27b: $tType,V0s: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1t: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( V0s = V1t )
<=> ! [V2x: A_27a,V3y: A_27b] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) @ V0s )
= ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V3y ) @ V1t ) ) ) ).
thf(thm_2Eset__relation_2Ein__domain,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: A_27a,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0x @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V1r ) )
<=> ? [V2y: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V0x @ V2y ) @ V1r ) ) ).
thf(thm_2Eset__relation_2Ein__range,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27a,V1r: ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0y @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b @ V1r ) )
<=> ? [V2x: A_27b] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) @ ( c_2Epair_2E_2C @ A_27b @ A_27a @ V2x @ V0y ) @ V1r ) ) ).
thf(thm_2Eset__relation_2Ein__dom__rg,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) @ V2r )
=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V2r ) )
& ( c_2Ebool_2EIN @ A_27b @ V0y @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a @ V2r ) ) ) ) ).
thf(thm_2Eset__relation_2Edomain__mono,axiom,
! [A_27a: $tType,A_27b: $tType,V0r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1r @ V0r_27 )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V1r ) @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V0r_27 ) ) ) ).
thf(thm_2Eset__relation_2Erange__mono,axiom,
! [A_27a: $tType,A_27b: $tType,V0r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1r @ V0r_27 )
=> ( c_2Epred__set_2ESUBSET @ A_27b @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a @ V1r ) @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a @ V0r_27 ) ) ) ).
thf(thm_2Eset__relation_2Ein__rrestrict,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V3s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V0x @ V1y ) @ ( c_2Eset__relation_2Errestrict @ A_27a @ V2r @ V3s ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V0x @ V1y ) @ V2r )
& ( c_2Ebool_2EIN @ A_27a @ V0x @ V3s )
& ( c_2Ebool_2EIN @ A_27a @ V1y @ V3s ) ) ) ).
thf(thm_2Eset__relation_2Ein__rrestrict__alt,axiom,
! [A_27a: $tType,V0x: tyop_2Epair_2Eprod @ A_27a @ A_27a,V1s: A_27a > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0x @ ( c_2Eset__relation_2Errestrict @ A_27a @ V2r @ V1s ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0x @ V2r )
& ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2EFST @ A_27a @ A_27a @ V0x ) @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ ( c_2Epair_2ESND @ A_27a @ A_27a @ V0x ) @ V1s ) ) ) ).
thf(thm_2Eset__relation_2Errestrict__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0s ) @ V1r ) ).
thf(thm_2Eset__relation_2Errestrict__union,axiom,
! [A_27a: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s: A_27a > $o] :
( ( c_2Eset__relation_2Errestrict @ A_27a @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r1 @ V1r2 ) @ V2s )
= ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r1 @ V2s ) @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r2 @ V2s ) ) ) ).
thf(thm_2Eset__relation_2Errestrict__rrestrict,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a > $o,V2y: A_27a > $o] :
( ( c_2Eset__relation_2Errestrict @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V1x ) @ V2y )
= ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ ( c_2Epred__set_2EINTER @ A_27a @ V1x @ V2y ) ) ) ).
thf(thm_2Eset__relation_2Edomain__rrestrict__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0s ) ) @ V0s ) ).
thf(thm_2Eset__relation_2Erange__rrestrict__SUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0s ) ) @ V0s ) ).
thf(thm_2Eset__relation_2Estrict__rrestrict,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Estrict @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V1s ) )
= ( c_2Eset__relation_2Errestrict @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) @ V1s ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__subset__s,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1s: A_27b > $o,V2s_27: A_27b > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27b @ V2s_27 @ V1s ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V2s_27 ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__subset__r,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V2s: A_27b > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V2s )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1r_27 @ V0r ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V1r_27 @ V2s ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__subset__rs,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1s: A_27b > $o,V2r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V3s_27: A_27b > $o] :
( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V1s )
=> ( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V2r_27 @ V0r )
=> ( ( c_2Epred__set_2ESUBSET @ A_27b @ V3s_27 @ V1s )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V2r_27 @ V3s_27 ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__subset,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2s_27: A_27a > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0r @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2s_27 @ V1s ) )
=> ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0r @ V2s_27 )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V2s_27 ) @ V2s_27 ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__union,axiom,
! [A_27a: $tType,A_27b: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V2s1: A_27b > $o,V3s2: A_27b > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r1 @ V2s1 )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V1r2 @ V3s2 ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0r1 @ V1r2 ) @ ( c_2Epred__set_2EINTER @ A_27b @ V2s1 @ V3s2 ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__comp,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) > $o,V2s1: A_27b > $o,V3s2: A_27c > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r1 @ V2s1 )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27c @ V1r2 @ V3s2 )
& ( c_2Epred__set_2ESUBSET @ A_27b
@ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b
@ ^ [V4x: A_27b] :
( c_2Epair_2E_2C @ A_27b @ $o @ V4x
@ ( c_2Ebool_2E_3F @ A_27c
@ ^ [V5y: A_27c] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2EIN @ A_27c @ V5y @ V3s2 ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27b @ A_27c ) @ ( c_2Epair_2E_2C @ A_27b @ A_27c @ V4x @ V5y ) @ V1r2 ) ) ) ) )
@ V2s1 ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27c @ ( c_2Eset__relation_2Ercomp @ A_27a @ A_27c @ A_27b @ V0r1 @ V1r2 ) @ V3s2 ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__inj__image,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: A_27a > A_27b,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s: A_27a > $o] :
( ( ! [V3x: A_27a,V4y: A_27a] :
( ( ( V0f @ V3x )
= ( V0f @ V4y ) )
=> ( V3x = V4y ) )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V1r @ V2s ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27b @ A_27b
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27b @ A_27b ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27b @ A_27b ) @ $o )
@ ^ [V5x: A_27a,V6y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27b @ A_27b ) @ $o @ ( c_2Epair_2E_2C @ A_27b @ A_27b @ ( V0f @ V5x ) @ ( V0f @ V6y ) ) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V5x @ V6y ) @ V1r ) ) ) )
@ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b @ V0f @ V2s ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefixes__range,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1s: A_27b > $o,V2t: A_27b > $o] :
( ( ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ V1s )
& ( c_2Epred__set_2EDISJOINT @ A_27b @ V2t @ ( c_2Eset__relation_2Erange @ A_27b @ A_27a @ V0r ) ) )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27b @ V0r @ ( c_2Epred__set_2EUNION @ A_27b @ V1s @ V2t ) ) ) ).
thf(thm_2Eset__relation_2Etc__rules,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ! [V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ? [V5z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V5z ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V5z @ V4y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3x @ V4y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__cases,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
| ? [V3z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3z ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3z @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__strongind,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V6z ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( V1tc_27 @ V4x @ V6z )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6z @ V5y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( V1tc_27 @ V6z @ V5y ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Etc__cases__right,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
| ? [V3z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3z ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3z @ V2y ) @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__cases__left,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
<=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ V0r )
| ? [V3z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V3z ) @ V0r )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3z @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__ind__left,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V6z ) @ V0r )
& ( V1tc_27 @ V6z @ V5y ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Etc__strongind__left,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V6z ) @ V0r )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6z @ V5y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( V1tc_27 @ V6z @ V5y ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Etc__ind__right,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( V1tc_27 @ V4x @ V6z )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6z @ V5y ) @ V0r ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Ertc__ind__right,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) )
| ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) ) )
=> ( V1tc_27 @ V2x @ V2x ) )
& ! [V3x: A_27a,V4y: A_27a] :
( ? [V5z: A_27a] :
( ( V1tc_27 @ V3x @ V5z )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V5z @ V4y ) @ V0r ) )
=> ( V1tc_27 @ V3x @ V4y ) ) )
=> ! [V6x: A_27a,V7y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6x @ V7y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V6x @ V7y ) ) ) ).
thf(thm_2Eset__relation_2Etc__strongind__right,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1tc_27: A_27a > A_27a > $o] :
( ( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
=> ( V1tc_27 @ V2x @ V3y ) )
& ! [V4x: A_27a,V5y: A_27a] :
( ? [V6z: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V6z ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
& ( V1tc_27 @ V4x @ V6z )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V6z @ V5y ) @ V0r ) )
=> ( V1tc_27 @ V4x @ V5y ) ) )
=> ! [V7x: A_27a,V8y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V7x @ V8y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( V1tc_27 @ V7x @ V8y ) ) ) ).
thf(thm_2Eset__relation_2Etc__union,axiom,
! [A_27a: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r1 ) )
=> ! [V3r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r1 @ V3r2 ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__implication,axiom,
! [A_27a: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ! [V2x: A_27a,V3y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r1 )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V1r2 ) )
=> ! [V4x: A_27a,V5y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V5y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r1 ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x @ V5y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r2 ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__empty,axiom,
! [A_27a: $tType,V0x: A_27a,V1y: A_27a] : ( (~) @ ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V0x @ V1y ) @ ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ).
thf(thm_2Eset__relation_2Etc__empty__eqn,axiom,
! [A_27a: $tType] :
( ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) ) )
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ).
thf(thm_2Eset__relation_2Etc__domain__range,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1x: A_27a,V2y: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V2y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) )
& ( c_2Ebool_2EIN @ A_27a @ V2y @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Errestrict__tc,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2e: A_27a,V3e_27: A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2e @ V3e_27 ) @ ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0x ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2e @ V3e_27 ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Etc__SWAP,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etc @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) @ V0r ) )
= ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__subset,axiom,
! [A_27a: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r1 )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V1r2 @ V0r1 ) )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r2 ) ) ).
thf(thm_2Eset__relation_2Eacyclic__union,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EDISJOINT @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0r ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0r ) ) @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r_27 ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r_27 ) ) )
& ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r )
& ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r_27 ) )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r @ V1r_27 ) ) ) ).
thf(thm_2Eset__relation_2Etransitive__tc,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
=> ( ( c_2Eset__relation_2Etc @ A_27a @ V0r )
= V0r ) ) ).
thf(thm_2Eset__relation_2Etc__transitive,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Eset__relation_2Etransitive @ A_27a @ ( c_2Eset__relation_2Etc @ A_27a @ V0r ) ) ).
thf(thm_2Eset__relation_2Eantisym__subset,axiom,
! [A_27a: $tType,V0t: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eantisym @ A_27a @ V0t )
=> ( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V1s @ V0t )
=> ( c_2Eset__relation_2Eantisym @ A_27a @ V1s ) ) ) ).
thf(thm_2Eset__relation_2Epartial__order__dom__rng,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
& ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) ) ) ).
thf(thm_2Eset__relation_2Epartial__order__subset,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2s_27: A_27a > $o] :
( ( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2s_27 @ V1s ) )
=> ( c_2Eset__relation_2Epartial__order @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V2s_27 ) @ V2s_27 ) ) ).
thf(thm_2Eset__relation_2Estrict__partial__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) ) @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) ) @ V1s )
& ( c_2Eset__relation_2Etransitive @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) )
& ( c_2Eset__relation_2Eantisym @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Estrict__partial__order__acyclic,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__subset,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2s_27: A_27a > $o] :
( ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2s_27 @ V1s ) )
=> ( c_2Eset__relation_2Elinear__order @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V0r @ V2s_27 ) @ V2s_27 ) ) ).
thf(thm_2Eset__relation_2Epartial__order__linear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s )
=> ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__dom__rng,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
& ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ V1s ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__dom__rng,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V0r )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3y @ V1s ) ) ) ).
thf(thm_2Eset__relation_2ERREFL__EXP__RSUBSET,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1R: A_27a > A_27a > $o] : ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ V1R @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ V1R @ V0s ) ) ).
thf(thm_2Eset__relation_2ERREFL__EXP__UNIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ V0R @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0R ) ).
thf(thm_2Eset__relation_2EREL__RESTRICT__UNIV,axiom,
! [A_27a: $tType,V0R: A_27a > A_27a > $o] :
( ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V0R @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V0R ) ).
thf(thm_2Eset__relation_2Ein__rel__to__reln,axiom,
! [A_27a: $tType,A_27b: $tType,V0xy: tyop_2Epair_2Eprod @ A_27a @ A_27b,V1R: A_27a > A_27b > $o] :
( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V0xy @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V1R ) )
= ( V1R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V0xy ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V0xy ) ) ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__app,axiom,
! [A_27a: $tType,A_27b: $tType,V0y: A_27b,V1x: A_27a,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V2r @ V1x @ V0y )
= ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V1x @ V0y ) @ V2r ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__IS__UNCURRY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b )
= ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ $o ) ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__IS__CURRY,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b )
= ( c_2Epair_2ECURRY @ A_27a @ A_27b @ $o ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__inv,axiom,
! [A_27a: $tType,A_27b: $tType,V0R: A_27a > A_27b > $o] :
( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V0R ) )
= V0R ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__inv,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r ) )
= V0r ) ).
thf(thm_2Eset__relation_2Ereln__to__rel__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V1r1 )
= ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r2 ) )
<=> ( V1r1 = V0r2 ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__11,axiom,
! [A_27a: $tType,A_27b: $tType,V0R2: A_27a > A_27b > $o,V1R1: A_27a > A_27b > $o] :
( ( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V1R1 )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V0R2 ) )
<=> ( V1R1 = V0R2 ) ) ).
thf(thm_2Eset__relation_2Erel__to__reln__swap,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V1R: A_27a > A_27b > $o] :
( ( V0r
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V1R ) )
<=> ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r )
= V1R ) ) ).
thf(thm_2Eset__relation_2Edomain__to__rel__conv,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o] :
( ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V0r )
= ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Erange__to__rel__conv,axiom,
! [A_27a: $tType,A_27b: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o] :
( ( c_2Eset__relation_2Erange @ A_27a @ A_27b @ V0r )
= ( c_2Erelation_2ERRANGE @ A_27b @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Estrict__to__rel__conv,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Estrict @ A_27a @ V0r )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Errestrict__to__rel__conv,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0s )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Ercomp__to__rel__conv,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0r2: ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) > $o,V1r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > $o] :
( ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c @ V1r1 @ V0r2 )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b @ V0r2 ) @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c @ V1r1 ) ) ) ) ).
thf(thm_2Eset__relation_2Euniv__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Eset__relation_2Euniv__reln @ A_27a @ V0s )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Eset__relation_2ERRUNIV @ A_27a @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Etc__to__rel__conv,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etc @ A_27a @ V0r )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ) ).
thf(thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eirreflexive @ A_27a @ V1r @ V0s )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Eirreflexive__reln__to__rel__conv__UNIV,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eirreflexive @ A_27a @ V0r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Ereflexive__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Ereflexive @ A_27a @ V1r @ V0s )
= ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) @ V0s ) ) ) ).
thf(thm_2Eset__relation_2Ereflexive__reln__to__rel__conv__UNIV,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Ereflexive @ A_27a @ V0r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Etransitive__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
= ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eantisym__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eantisym @ A_27a @ V0r )
= ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Epartial__order__reln__to__rel__conv,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Epartial__order @ A_27a @ V1r @ V0s )
<=> ( ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) @ ( c_2Eset__relation_2ERRUNIV @ A_27a @ V0s ) )
& ( c_2Erelation_2EWeakOrder @ A_27a @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) @ V0s ) ) ) ) ).
thf(thm_2Eset__relation_2Epartial__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EWeakOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EWeakLinearOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__reln__to__rel__conv__UNIV,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Ereln__rel__conv__thms,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0y: A_27b,V1xy: tyop_2Epair_2Eprod @ A_27a @ A_27b,V2x: A_27a,V3s: A_27a > $o,V4r2: ( tyop_2Epair_2Eprod @ A_27c @ A_27b ) > $o,V5r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V6r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) > $o,V7r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V8r: ( tyop_2Epair_2Eprod @ A_27b @ A_27a ) > $o,V9r: ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > $o,V10r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V11R2: A_27a > A_27b > $o,V12R1: A_27a > A_27b > $o,V13R: A_27a > A_27b > $o,V14R: A_27a > A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ V1xy @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V13R ) )
= ( V13R @ ( c_2Epair_2EFST @ A_27a @ A_27b @ V1xy ) @ ( c_2Epair_2ESND @ A_27a @ A_27b @ V1xy ) ) )
& ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V9r @ V2x @ V0y )
= ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) @ ( c_2Epair_2E_2C @ A_27a @ A_27b @ V2x @ V0y ) @ V9r ) )
& ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V13R ) )
= V13R )
& ( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V9r ) )
= V9r )
& ( ( ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V7r1 )
= ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V5r2 ) )
<=> ( V7r1 = V5r2 ) )
& ( ( ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V12R1 )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ V11R2 ) )
<=> ( V12R1 = V11R2 ) )
& ( ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ V14R @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V14R )
& ( ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ V14R @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= V14R )
& ( ( c_2Eset__relation_2Edomain @ A_27a @ A_27b @ V9r )
= ( c_2Erelation_2ERDOM @ A_27a @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27b @ V9r ) ) )
& ( ( c_2Eset__relation_2Erange @ A_27a @ A_27b @ V8r )
= ( c_2Erelation_2ERRANGE @ A_27b @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27b @ A_27a @ V8r ) ) )
& ( ( c_2Eset__relation_2Estrict @ A_27a @ V10r )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Erelation_2ESTRORD @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) ) )
& ( ( c_2Eset__relation_2Errestrict @ A_27a @ V10r @ V3s )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) @ V3s ) ) )
& ( ( c_2Eset__relation_2Ercomp @ A_27a @ A_27b @ A_27c @ V6r1 @ V4r2 )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27b @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b @ ( c_2Eset__relation_2Ereln__to__rel @ A_27c @ A_27b @ V4r2 ) @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27c @ V6r1 ) ) ) )
& ( ( c_2Eset__relation_2Euniv__reln @ A_27a @ V3s )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Eset__relation_2ERRUNIV @ A_27a @ V3s ) ) )
& ( ( c_2Eset__relation_2Etc @ A_27a @ V10r )
= ( c_2Eset__relation_2Erel__to__reln @ A_27a @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) ) )
& ( ( c_2Eset__relation_2Eacyclic @ A_27a @ V10r )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Erelation_2ETC @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) ) )
& ( ( c_2Eset__relation_2Eirreflexive @ A_27a @ V10r @ V3s )
= ( c_2Erelation_2Eirreflexive @ A_27a @ ( c_2Epred__set_2EREL__RESTRICT @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) @ V3s ) ) )
& ( ( c_2Eset__relation_2Ereflexive @ A_27a @ V10r @ V3s )
= ( c_2Erelation_2Ereflexive @ A_27a @ ( c_2Eset__relation_2ERREFL__EXP @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) @ V3s ) ) )
& ( ( c_2Eset__relation_2Etransitive @ A_27a @ V10r )
= ( c_2Erelation_2Etransitive @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) )
& ( ( c_2Eset__relation_2Eantisym @ A_27a @ V10r )
= ( c_2Erelation_2Eantisymmetric @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) )
& ( ( c_2Eset__relation_2Epartial__order @ A_27a @ V10r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EWeakOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) )
& ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V10r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EWeakLinearOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) )
& ( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V10r @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Erelation_2EStrongLinearOrder @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V10r ) ) ) ) ).
thf(thm_2Eset__relation_2Eacyclic__WF,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r ) @ V0s ) )
=> ( c_2Erelation_2EWF @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__elements__subset,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1lo ) @ V0s ) ).
thf(thm_2Eset__relation_2Eminimal__elements__SWAP,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2ESWAP @ A_27a @ A_27a ) @ V1r ) )
= ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0xs @ V1r ) ) ).
thf(thm_2Eset__relation_2Emaximal__union,axiom,
! [A_27a: $tType,V0e: A_27a,V1s: A_27a > $o,V2r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V3r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V1s @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V2r1 @ V3r2 ) ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V1s @ V2r1 ) )
& ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V1s @ V3r2 ) ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__union,axiom,
! [A_27a: $tType,V0e: A_27a,V1s: A_27a > $o,V2r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V3r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V1s @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V2r1 @ V3r2 ) ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V1s @ V2r1 ) )
& ( c_2Ebool_2EIN @ A_27a @ V0e @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V1s @ V3r2 ) ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__elements__mono,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V2r @ V1r_27 )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ V1r_27 ) @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ V2r ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__elements__rrestrict,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V0xs ) )
= ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0xs @ V1r ) ) ).
thf(thm_2Eset__relation_2EWF__has__minimal__path,axiom,
! [A_27a: $tType,V0x: A_27a,V1s: A_27a > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Erelation_2EWF @ A_27a @ ( c_2Eset__relation_2Ereln__to__rel @ A_27a @ A_27a @ V2r ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V1s )
=> ? [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V1s @ V2r ) )
& ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V0x ) @ ( c_2Eset__relation_2Etc @ A_27a @ V2r ) )
| ( V3y = V0x ) ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__acyclic__has__maximal,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ! [V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__acyclic__has__minimal,axiom,
! [A_27a: $tType,V0s: A_27a > $o] :
( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
=> ( ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
=> ! [V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Emaximal__TC,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r ) @ V0s ) )
=> ( ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) )
= ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Eminimal__TC,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r ) @ V0s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r ) @ V0s ) )
=> ( ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) )
= ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Efinite__acyclic__has__minimal__path,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2x: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) ) )
=> ? [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V2x ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__acyclic__has__maximal__path,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2x: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Eacyclic @ A_27a @ V1r )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V0s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) ) )
=> ? [V3y: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ ( c_2Eset__relation_2Etc @ A_27a @ V1r ) ) ) ) ).
thf(thm_2Eset__relation_2Efinite__prefix__po__has__minimal__path,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x: A_27a,V3s_27: A_27a > $o] :
( ( ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0r @ V1s )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V3s_27 @ V0r ) ) )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V3s_27 )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3s_27 @ V1s ) )
=> ? [V4x_27: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x_27 @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V3s_27 @ V0r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V4x_27 @ V2x ) @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eempty__strict__linear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0r
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ).
thf(thm_2Eset__relation_2Eempty__linear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0r
= ( c_2Epred__set_2EEMPTY @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__restrict,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s_27: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V0s )
=> ( c_2Eset__relation_2Elinear__order @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V2s_27 ) @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V2s_27 ) ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__restrict,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s_27: A_27a > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V1r @ V0s )
=> ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ ( c_2Eset__relation_2Errestrict @ A_27a @ V1r @ V2s_27 ) @ ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V2s_27 ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__dom__rg,axiom,
! [A_27a: $tType,V0lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1X: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0lo @ V1X )
=> ( ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V0lo ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V0lo ) )
= V1X ) ) ).
thf(thm_2Eset__relation_2Elinear__order__refl,axiom,
! [A_27a: $tType,V0x: A_27a,V1lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2X: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V1lo @ V2X )
=> ( ( c_2Ebool_2EIN @ A_27a @ V0x @ V2X )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V0x @ V0x ) @ V1lo ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__in__set,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V3X: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V2lo @ V3X )
=> ( ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V1x @ V0y ) @ V2lo )
=> ( ( c_2Ebool_2EIN @ A_27a @ V1x @ V3X )
& ( c_2Ebool_2EIN @ A_27a @ V0y @ V3X ) ) ) ) ).
thf(thm_2Eset__relation_2EIN__MIN__LO,axiom,
! [A_27a: $tType,V0y: A_27a,V1x: A_27a,V2lo: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V3X: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V1x @ V3X )
=> ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V2lo @ V3X )
=> ( ( c_2Ebool_2EIN @ A_27a @ V0y @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V3X @ V2lo ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V0y @ V1x ) @ V2lo ) ) ) ) ).
thf(thm_2Eset__relation_2Eextend__linear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x: A_27a] :
( ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s ) )
=> ( c_2Eset__relation_2Elinear__order @ A_27a
@ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a
@ ^ [V3y: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V2x ) @ ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) )
@ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Epred__set_2EINSERT @ A_27a @ V2x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__acyclic,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ V1s )
=> ( c_2Eset__relation_2Eacyclic @ A_27a @ V0r ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order__union__acyclic,axiom,
! [A_27a: $tType,V0r1: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1r2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s: A_27a > $o] :
( ( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r1 @ V2s )
& ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a @ V1r2 ) @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a @ V1r2 ) ) @ V2s ) )
=> ( ( c_2Eset__relation_2Eacyclic @ A_27a @ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r1 @ V1r2 ) )
= ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V1r2 @ V0r1 ) ) ) ).
thf(thm_2Eset__relation_2Estrict__linear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s )
=> ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ ( c_2Eset__relation_2Estrict @ A_27a @ V0r ) @ V1s ) ) ).
thf(thm_2Eset__relation_2Elinear__order,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V0r @ V1s )
=> ( c_2Eset__relation_2Elinear__order @ A_27a
@ ( c_2Epred__set_2EUNION @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a
@ ^ [V2x: A_27a] : ( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V2x ) @ ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s ) ) ) )
@ V1s ) ) ).
thf(thm_2Eset__relation_2Efinite__strict__linear__order__has__maximal,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V1r @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Efinite__strict__linear__order__has__minimal,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Estrict__linear__order @ A_27a @ V1r @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Efinite__linear__order__has__maximal,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Efinite__linear__order__has__minimal,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V0s )
& ( (~)
@ ( V0s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) )
=> ? [V2x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Emaximal__linear__order,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3y @ V0s )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V0s @ V1r ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V3y @ V2x ) @ V1r ) ) ).
thf(thm_2Eset__relation_2Eminimal__linear__order,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2x: A_27a,V3y: A_27a] :
( ( ( c_2Ebool_2EIN @ A_27a @ V3y @ V0s )
& ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V0s )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V0s @ V1r ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x @ V3y ) @ V1r ) ) ).
thf(thm_2Eset__relation_2Eminimal__linear__order__unique,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2s_27: A_27a > $o,V3x: A_27a,V4y: A_27a] :
( ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V0r @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V2s_27 @ V0r ) )
& ( c_2Ebool_2EIN @ A_27a @ V4y @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V2s_27 @ V0r ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2s_27 @ V1s ) )
=> ( V3x = V4y ) ) ).
thf(thm_2Eset__relation_2Efinite__prefix__linear__order__has__unique__minimal,axiom,
! [A_27a: $tType,V0x: A_27a,V1r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s: A_27a > $o,V3s_27: A_27a > $o] :
( ( ( c_2Eset__relation_2Elinear__order @ A_27a @ V1r @ V2s )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V1r @ V2s )
& ( c_2Ebool_2EIN @ A_27a @ V0x @ V3s_27 )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3s_27 @ V2s ) )
=> ( c_2Epred__set_2ESING @ A_27a @ ( c_2Eset__relation_2Eminimal__elements @ A_27a @ V3s_27 @ V1r ) ) ) ).
thf(thm_2Eset__relation_2Enat__order__iso__thm,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eoption_2Eoption @ A_27a ),V1s: A_27a > $o] :
( ( ! [V2n: tyop_2Enum_2Enum,V3m: tyop_2Enum_2Enum] :
( ( ( ( V0f @ V3m )
= ( V0f @ V2n ) )
& ( (~)
@ ( ( V0f @ V3m )
= ( c_2Eoption_2ENONE @ A_27a ) ) ) )
=> ( V3m = V2n ) )
& ! [V4x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V4x @ V1s )
=> ? [V5m: tyop_2Enum_2Enum] :
( ( V0f @ V5m )
= ( c_2Eoption_2ESOME @ A_27a @ V4x ) ) )
& ! [V6m: tyop_2Enum_2Enum,V7x: A_27a] :
( ( ( V0f @ V6m )
= ( c_2Eoption_2ESOME @ A_27a @ V7x ) )
=> ( c_2Ebool_2EIN @ A_27a @ V7x @ V1s ) ) )
=> ( ( c_2Eset__relation_2Elinear__order @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V8x: A_27a,V9y: A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V8x @ V9y )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V10m: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V11n: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Earithmetic_2E_3C_3D @ V10m @ V11n ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V0f @ V10m ) @ ( c_2Eoption_2ESOME @ A_27a @ V8x ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V0f @ V11n ) @ ( c_2Eoption_2ESOME @ A_27a @ V9y ) ) ) ) ) ) ) ) )
@ V1s )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a )
@ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( tyop_2Epair_2Eprod @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o )
@ ^ [V12x: A_27a,V13y: A_27a] :
( c_2Epair_2E_2C @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ $o @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V12x @ V13y )
@ ( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V14m: tyop_2Enum_2Enum] :
( c_2Ebool_2E_3F @ tyop_2Enum_2Enum
@ ^ [V15n: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Earithmetic_2E_3C_3D @ V14m @ V15n ) @ ( c_2Ebool_2E_2F_5C @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V0f @ V14m ) @ ( c_2Eoption_2ESOME @ A_27a @ V12x ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( V0f @ V15n ) @ ( c_2Eoption_2ESOME @ A_27a @ V13y ) ) ) ) ) ) ) ) )
@ V1s ) ) ) ).
thf(thm_2Eset__relation_2Eupper__bounds__lem,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o,V2x1: A_27a,V3x2: A_27a] :
( ( ( c_2Eset__relation_2Etransitive @ A_27a @ V0r )
& ( c_2Ebool_2EIN @ A_27a @ V2x1 @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V1s @ V0r ) )
& ( c_2Ebool_2EIN @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ ( c_2Epair_2E_2C @ A_27a @ A_27a @ V2x1 @ V3x2 ) @ V0r ) )
=> ( c_2Ebool_2EIN @ A_27a @ V3x2 @ ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V1s @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Ezorns__lemma,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
& ! [V2t: A_27a > $o] :
( ( c_2Eset__relation_2Echain @ A_27a @ V2t @ V0r )
=> ( (~)
@ ( ( c_2Eset__relation_2Eupper__bounds @ A_27a @ A_27a @ V2t @ V0r )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) )
=> ? [V3x: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Eset__relation_2Emaximal__elements @ A_27a @ V1s @ V0r ) ) ) ).
thf(thm_2Eset__relation_2Eper__delete,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1xss: ( A_27a > $o ) > $o,V2e: A_27a] :
( ( c_2Eset__relation_2Eper @ A_27a @ V0xs @ V1xss )
=> ( c_2Eset__relation_2Eper @ A_27a @ ( c_2Epred__set_2EDELETE @ A_27a @ V0xs @ V2e )
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V3es: A_27a > $o] :
( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V3es
@ ( c_2Ebool_2E_2F_5C
@ ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3es
@ ( c_2Epred__set_2EIMAGE @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V4es: A_27a > $o] : ( c_2Epred__set_2EDELETE @ A_27a @ V4es @ V2e )
@ V1xss ) )
@ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( A_27a > $o ) @ V3es @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ) ).
thf(thm_2Eset__relation_2Eper__restrict__per,axiom,
! [A_27a: $tType,V0r: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2s_27: A_27a > $o] :
( ( c_2Eset__relation_2Eper @ A_27a @ V1s @ V0r )
=> ( c_2Eset__relation_2Eper @ A_27a @ V2s_27 @ ( c_2Eset__relation_2Eper__restrict @ A_27a @ V0r @ V2s_27 ) ) ) ).
thf(thm_2Eset__relation_2Ecountable__per,axiom,
! [A_27a: $tType,V0xs: A_27a > $o,V1xss: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V0xs )
& ( c_2Eset__relation_2Eper @ A_27a @ V0xs @ V1xss ) )
=> ( c_2Epred__set_2Ecountable @ ( A_27a > $o ) @ V1xss ) ) ).
thf(thm_2Eset__relation_2Eall__choices__thm,axiom,
! [A_27a: $tType,V0x: A_27a > $o,V1s: ( A_27a > $o ) > $o,V2y: A_27a] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V0x @ ( c_2Eset__relation_2Eall__choices @ A_27a @ V1s ) )
& ( c_2Ebool_2EIN @ A_27a @ V2y @ V0x ) )
=> ? [V3z: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3z @ V1s )
& ( c_2Ebool_2EIN @ A_27a @ V2y @ V3z ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__num__order,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ tyop_2Enum_2Enum @ V0f @ V1s @ V2t )
=> ( c_2Eset__relation_2Elinear__order @ A_27a @ ( c_2Eset__relation_2Enum__order @ A_27a @ V0f @ V1s ) @ V1s ) ) ).
thf(thm_2Eset__relation_2Enum__order__finite__prefix,axiom,
! [A_27a: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1s: A_27a > $o,V2t: tyop_2Enum_2Enum > $o] :
( ( c_2Epred__set_2EINJ @ A_27a @ tyop_2Enum_2Enum @ V0f @ V1s @ V2t )
=> ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ ( c_2Eset__relation_2Enum__order @ A_27a @ V0f @ V1s ) @ V1s ) ) ).
thf(thm_2Eset__relation_2Enth__min__ind,axiom,
! [A_27a: $tType,V0P: ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) ) > tyop_2Enum_2Enum > $o] :
( ( ! [V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2s: A_27a > $o,V3r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] : ( V0P @ V1r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V2s @ V3r ) @ c_2Enum_2E0 )
& ! [V4r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V5s: A_27a > $o,V6r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V7n: tyop_2Enum_2Enum] :
( ! [V8min: tyop_2Eoption_2Eoption @ A_27a] :
( ( ( V8min
= ( c_2Eset__relation_2Eget__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V5s @ V6r ) ) )
& ( (~)
@ ( V8min
= ( c_2Eoption_2ENONE @ A_27a ) ) ) )
=> ( V0P @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V5s @ ( c_2Eoption_2ETHE @ A_27a @ V8min ) ) @ V6r ) @ V7n ) )
=> ( V0P @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V5s @ V6r ) @ ( c_2Enum_2ESUC @ V7n ) ) ) )
=> ! [V9v: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V10v1: A_27a > $o,V11v2: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V12v3: tyop_2Enum_2Enum] : ( V0P @ V9v @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V10v1 @ V11v2 ) @ V12v3 ) ) ).
thf(thm_2Eset__relation_2Enth__min__def,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o,V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Enth__min @ A_27a @ V1r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V0s @ V2r ) @ c_2Enum_2E0 )
= ( c_2Eset__relation_2Eget__min @ A_27a @ V1r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V0s @ V2r ) ) )
& ! [V3s: A_27a > $o,V4r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V5r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V6n: tyop_2Enum_2Enum] :
( ( c_2Eset__relation_2Enth__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V3s @ V5r ) @ ( c_2Enum_2ESUC @ V6n ) )
= ( c_2Ebool_2ELET @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V7min: tyop_2Eoption_2Eoption @ A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V7min @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eset__relation_2Enth__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V3s @ ( c_2Eoption_2ETHE @ A_27a @ V7min ) ) @ V5r ) @ V6n ) )
@ ( c_2Eset__relation_2Eget__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V3s @ V5r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Enth__min__def__compute,axiom,
! [A_27a: $tType] :
( ! [V0s: A_27a > $o,V1r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V2r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Enth__min @ A_27a @ V1r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V0s @ V2r ) @ c_2Enum_2E0 )
= ( c_2Eset__relation_2Eget__min @ A_27a @ V1r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V0s @ V2r ) ) )
& ! [V3s: A_27a > $o,V4r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V5r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V6n: tyop_2Enum_2Enum] :
( ( c_2Eset__relation_2Enth__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V3s @ V5r ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6n ) ) )
= ( c_2Ebool_2ELET @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V7min: tyop_2Eoption_2Eoption @ A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V7min @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eset__relation_2Enth__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V3s @ ( c_2Eoption_2ETHE @ A_27a @ V7min ) ) @ V5r ) @ ( c_2Earithmetic_2E_2D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V6n ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
@ ( c_2Eset__relation_2Eget__min @ A_27a @ V4r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V3s @ V5r ) ) ) )
& ! [V8s: A_27a > $o,V9r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V10r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V11n: tyop_2Enum_2Enum] :
( ( c_2Eset__relation_2Enth__min @ A_27a @ V9r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V8s @ V10r ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ V11n ) ) )
= ( c_2Ebool_2ELET @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( tyop_2Eoption_2Eoption @ A_27a )
@ ^ [V12min: tyop_2Eoption_2Eoption @ A_27a] : ( c_2Ebool_2ECOND @ ( tyop_2Eoption_2Eoption @ A_27a ) @ ( c_2Emin_2E_3D @ ( tyop_2Eoption_2Eoption @ A_27a ) @ V12min @ ( c_2Eoption_2ENONE @ A_27a ) ) @ ( c_2Eoption_2ENONE @ A_27a ) @ ( c_2Eset__relation_2Enth__min @ A_27a @ V9r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ ( c_2Epred__set_2EDELETE @ A_27a @ V8s @ ( c_2Eoption_2ETHE @ A_27a @ V12min ) ) @ V10r ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ V11n ) ) ) )
@ ( c_2Eset__relation_2Eget__min @ A_27a @ V9r_27 @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ ( ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o ) @ V8s @ V10r ) ) ) ) ) ).
thf(thm_2Eset__relation_2Elinear__order__of__countable__po,axiom,
! [A_27a: $tType,V0r: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o,V1s: A_27a > $o] :
( ( ( c_2Epred__set_2Ecountable @ A_27a @ V1s )
& ( c_2Eset__relation_2Epartial__order @ A_27a @ V0r @ V1s )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V0r @ V1s ) )
=> ? [V2r_27: ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) > $o] :
( ( c_2Eset__relation_2Elinear__order @ A_27a @ V2r_27 @ V1s )
& ( c_2Eset__relation_2Efinite__prefixes @ A_27a @ A_27a @ V2r_27 @ V1s )
& ( c_2Epred__set_2ESUBSET @ ( tyop_2Epair_2Eprod @ A_27a @ A_27a ) @ V0r @ V2r_27 ) ) ) ).
%------------------------------------------------------------------------------