ITP001 Axioms: ITP011^5.ax
%------------------------------------------------------------------------------
% File : ITP011^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : relation^2.ax [Gau20]
% : HL4011^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 346 ( 1 unt; 51 typ; 0 def)
% Number of atoms : 4405 ( 93 equ; 0 cnn)
% Maximal formula atoms : 61 ( 12 avg)
% Number of connectives : 9168 ( 11 ~; 16 |; 162 &;7935 @)
% ( 92 <=>; 952 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 14 avg;7935 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 71 ( 71 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 68 usr; 19 con; 0-3 aty)
% Number of variables : 1127 ( 10 ^1098 !; 19 ?;1127 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Erelation_2ECR,type,
c_2Erelation_2ECR: del > $i ).
thf(mem_c_2Erelation_2ECR,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ECR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EEMPTY__REL,type,
c_2Erelation_2EEMPTY__REL: del > $i ).
thf(mem_c_2Erelation_2EEMPTY__REL,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Erelation_2EEQC,type,
c_2Erelation_2EEQC: del > $i ).
thf(mem_c_2Erelation_2EEQC,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EEQC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2EIDEM,type,
c_2Erelation_2EIDEM: del > $i ).
thf(mem_c_2Erelation_2EIDEM,axiom,
! [A_27z: del] : ( mem @ ( c_2Erelation_2EIDEM @ A_27z ) @ ( arr @ ( arr @ A_27z @ A_27z ) @ bool ) ) ).
thf(tp_c_2Erelation_2EINDUCTIVE__INVARIANT,type,
c_2Erelation_2EINDUCTIVE__INVARIANT: del > del > $i ).
thf(mem_c_2Erelation_2EINDUCTIVE__INVARIANT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2EINDUCTIVE__INVARIANT__ON,type,
c_2Erelation_2EINDUCTIVE__INVARIANT__ON: del > del > $i ).
thf(mem_c_2Erelation_2EINDUCTIVE__INVARIANT__ON,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2EINVOL,type,
c_2Erelation_2EINVOL: del > $i ).
thf(mem_c_2Erelation_2EINVOL,axiom,
! [A_27z: del] : ( mem @ ( c_2Erelation_2EINVOL @ A_27z ) @ ( arr @ ( arr @ A_27z @ A_27z ) @ bool ) ) ).
thf(tp_c_2Erelation_2ELinearOrder,type,
c_2Erelation_2ELinearOrder: del > $i ).
thf(mem_c_2Erelation_2ELinearOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ELinearOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EO,type,
c_2Erelation_2EO: del > del > del > $i ).
thf(mem_c_2Erelation_2EO,axiom,
! [A_27g: del,A_27h: del,A_27k: del] : ( mem @ ( c_2Erelation_2EO @ A_27g @ A_27h @ A_27k ) @ ( arr @ ( arr @ A_27h @ ( arr @ A_27k @ bool ) ) @ ( arr @ ( arr @ A_27g @ ( arr @ A_27h @ bool ) ) @ ( arr @ A_27g @ ( arr @ A_27k @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2EOrder,type,
c_2Erelation_2EOrder: del > $i ).
thf(mem_c_2Erelation_2EOrder,axiom,
! [A_27g: del] : ( mem @ ( c_2Erelation_2EOrder @ A_27g ) @ ( arr @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EPreOrder,type,
c_2Erelation_2EPreOrder: del > $i ).
thf(mem_c_2Erelation_2EPreOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EPreOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2ERC,type,
c_2Erelation_2ERC: del > $i ).
thf(mem_c_2Erelation_2ERC,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ERC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2ERCOMPL,type,
c_2Erelation_2ERCOMPL: del > del > $i ).
thf(mem_c_2Erelation_2ERCOMPL,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERCOMPL @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2ERDOM,type,
c_2Erelation_2ERDOM: del > del > $i ).
thf(mem_c_2Erelation_2ERDOM,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Erelation_2ERDOM__DELETE,type,
c_2Erelation_2ERDOM__DELETE: del > del > $i ).
thf(mem_c_2Erelation_2ERDOM__DELETE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERDOM__DELETE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2ERESTRICT,type,
c_2Erelation_2ERESTRICT: del > del > $i ).
thf(mem_c_2Erelation_2ERESTRICT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) ) ) ) ).
thf(tp_c_2Erelation_2ERINTER,type,
c_2Erelation_2ERINTER: del > del > $i ).
thf(mem_c_2Erelation_2ERINTER,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2ERRANGE,type,
c_2Erelation_2ERRANGE: del > del > $i ).
thf(mem_c_2Erelation_2ERRANGE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERRANGE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27b @ bool ) ) ) ).
thf(tp_c_2Erelation_2ERRESTRICT,type,
c_2Erelation_2ERRESTRICT: del > del > $i ).
thf(mem_c_2Erelation_2ERRESTRICT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERRESTRICT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2ERSUBSET,type,
c_2Erelation_2ERSUBSET: del > del > $i ).
thf(mem_c_2Erelation_2ERSUBSET,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ bool ) ) ) ).
thf(tp_c_2Erelation_2ERTC,type,
c_2Erelation_2ERTC: del > $i ).
thf(mem_c_2Erelation_2ERTC,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ERTC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2ERUNION,type,
c_2Erelation_2ERUNION: del > del > $i ).
thf(mem_c_2Erelation_2ERUNION,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2ERUNIV,type,
c_2Erelation_2ERUNIV: del > del > $i ).
thf(mem_c_2Erelation_2ERUNIV,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) ).
thf(tp_c_2Erelation_2ESC,type,
c_2Erelation_2ESC: del > $i ).
thf(mem_c_2Erelation_2ESC,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ESC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2ESN,type,
c_2Erelation_2ESN: del > $i ).
thf(mem_c_2Erelation_2ESN,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ESN @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2ESTRORD,type,
c_2Erelation_2ESTRORD: del > $i ).
thf(mem_c_2Erelation_2ESTRORD,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2EStrongLinearOrder,type,
c_2Erelation_2EStrongLinearOrder: del > $i ).
thf(mem_c_2Erelation_2EStrongLinearOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EStrongOrder,type,
c_2Erelation_2EStrongOrder: del > $i ).
thf(mem_c_2Erelation_2EStrongOrder,axiom,
! [A_27g: del] : ( mem @ ( c_2Erelation_2EStrongOrder @ A_27g ) @ ( arr @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2ETC,type,
c_2Erelation_2ETC: del > $i ).
thf(mem_c_2Erelation_2ETC,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2ETC @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2EWCR,type,
c_2Erelation_2EWCR: del > $i ).
thf(mem_c_2Erelation_2EWCR,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EWCR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EWF,type,
c_2Erelation_2EWF: del > $i ).
thf(mem_c_2Erelation_2EWF,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EWF @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EWFP,type,
c_2Erelation_2EWFP: del > $i ).
thf(mem_c_2Erelation_2EWFP,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EWFP @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Erelation_2EWFREC,type,
c_2Erelation_2EWFREC: del > del > $i ).
thf(mem_c_2Erelation_2EWFREC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ) ).
thf(tp_c_2Erelation_2EWeakLinearOrder,type,
c_2Erelation_2EWeakLinearOrder: del > $i ).
thf(mem_c_2Erelation_2EWeakLinearOrder,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2EWeakOrder,type,
c_2Erelation_2EWeakOrder: del > $i ).
thf(mem_c_2Erelation_2EWeakOrder,axiom,
! [A_27g: del] : ( mem @ ( c_2Erelation_2EWeakOrder @ A_27g ) @ ( arr @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Eantisymmetric,type,
c_2Erelation_2Eantisymmetric: del > $i ).
thf(mem_c_2Erelation_2Eantisymmetric,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Eapprox,type,
c_2Erelation_2Eapprox: del > del > $i ).
thf(mem_c_2Erelation_2Eapprox,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2Eapprox @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2Ediag,type,
c_2Erelation_2Ediag: del > $i ).
thf(mem_c_2Erelation_2Ediag,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Ediag @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2Ediamond,type,
c_2Erelation_2Ediamond: del > $i ).
thf(mem_c_2Erelation_2Ediamond,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Ediamond @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Eequivalence,type,
c_2Erelation_2Eequivalence: del > $i ).
thf(mem_c_2Erelation_2Eequivalence,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Eequivalence @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Einv,type,
c_2Erelation_2Einv: del > del > $i ).
thf(mem_c_2Erelation_2Einv,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2Einv @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Erelation_2Einv__image,type,
c_2Erelation_2Einv__image: del > del > $i ).
thf(mem_c_2Erelation_2Einv__image,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2Einv__image @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Erelation_2Eirreflexive,type,
c_2Erelation_2Eirreflexive: del > $i ).
thf(mem_c_2Erelation_2Eirreflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Enf,type,
c_2Erelation_2Enf: del > del > $i ).
thf(mem_c_2Erelation_2Enf,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2Enf @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Erelation_2Ercdiamond,type,
c_2Erelation_2Ercdiamond: del > $i ).
thf(mem_c_2Erelation_2Ercdiamond,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Ercdiamond @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Ereflexive,type,
c_2Erelation_2Ereflexive: del > $i ).
thf(mem_c_2Erelation_2Ereflexive,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Esymmetric,type,
c_2Erelation_2Esymmetric: del > $i ).
thf(mem_c_2Erelation_2Esymmetric,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Ethe__fun,type,
c_2Erelation_2Ethe__fun: del > del > $i ).
thf(mem_c_2Erelation_2Ethe__fun,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Erelation_2Ethe__fun @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) ) ) ) ).
thf(tp_c_2Erelation_2Etotal,type,
c_2Erelation_2Etotal: del > $i ).
thf(mem_c_2Erelation_2Etotal,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Etotal @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Etransitive,type,
c_2Erelation_2Etransitive: del > $i ).
thf(mem_c_2Erelation_2Etransitive,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Erelation_2Etrichotomous,type,
c_2Erelation_2Etrichotomous: del > $i ).
thf(mem_c_2Erelation_2Etrichotomous,axiom,
! [A_27a: del] : ( mem @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(ax_thm_2Erelation_2Etransitive__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Ereflexive__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Eirreflexive__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ~ ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Esymmetric__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Eantisymmetric__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) )
=> ( V1x = V2y ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Eequivalence__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eequivalence @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Etotal__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
| ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Etrichotomous,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) )
<=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b ) )
| ( p @ ( ap @ ( ap @ V0R @ V2b ) @ V1a ) )
| ( V1a = V2b ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ETC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1a ) @ V2b ) )
<=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V3P @ V4x ) @ V5y ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V3P @ V6x ) @ V7y ) )
& ( p @ ( ap @ ( ap @ V3P @ V7y ) @ V8z ) ) )
=> ( p @ ( ap @ ( ap @ V3P @ V6x ) @ V8z ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V3P @ V1a ) @ V2b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERTC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1a ) @ V2b ) )
<=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ V3P @ V4x ) @ V4x ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ! [V7z: $i] :
( ( mem @ V7z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V5x ) @ V6y ) )
& ( p @ ( ap @ ( ap @ V3P @ V6y ) @ V7z ) ) )
=> ( p @ ( ap @ ( ap @ V3P @ V5x ) @ V7z ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V3P @ V1a ) @ V2b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( ( V1x = V2y )
| ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ESC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
| ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EEQC__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESC__SYMMETRIC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__TRANSITIVE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V1P @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V5z ) ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V6x ) @ V7y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__RULES,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V3x ) @ V5z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__RULES,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V1x ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3y ) @ V4z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2x ) @ V4z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__REFL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V1x ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__SINGLE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__STRONG__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V4y ) @ V5z ) )
& ( p @ ( ap @ ( ap @ V1P @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V5z ) ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V6x ) @ V7y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2y ) @ V3z ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V3z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__TRANSITIVE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__RTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__REFLEXIVE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__RTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__REFLEXIVE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V1Q ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V2x ) )
& ( p @ ( ap @ ( ap @ V1R @ V2x ) @ V3y ) ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V1R ) @ V4x ) @ V5y ) ) )
=> ( p @ ( ap @ V0P @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V1Q ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Eantisymmetric__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V1P @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7u: $i] :
( ( mem @ V7u @ A_27a )
=> ! [V8v: $i] :
( ( mem @ V8v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7u ) @ V8v ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7u ) @ V8v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__INDUCT__LEFT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V1P @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7x ) @ V8y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V0R @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7x ) @ V8y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V8y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__STRONG__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V1P @ V5y ) @ V6z ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7u: $i] :
( ( mem @ V7u @ A_27a )
=> ! [V8v: $i] :
( ( mem @ V8v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7u ) @ V8v ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7u ) @ V8v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__STRONG__INDUCT__LEFT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V1P @ V5y ) @ V6z ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7u: $i] :
( ( mem @ V7u @ A_27a )
=> ! [V8v: $i] :
( ( mem @ V8v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7u ) @ V8v ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7u ) @ V8v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
& ( p @ ( ap @ ( ap @ V0R @ V5y ) @ V6z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V6z ) ) ) ) ) ) )
=> ! [V7u: $i] :
( ( mem @ V7u @ A_27a )
=> ! [V8v: $i] :
( ( mem @ V8v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V7u ) @ V8v ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7u ) @ V8v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__INDUCT__ALT__LEFT,axiom,
! [A_27a: del,V0b: $i] :
( ( mem @ V0b @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1R @ V3x ) @ V0b ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1R @ V4x ) @ V5y ) )
& ( p @ ( ap @ V2Q @ V5y ) ) )
=> ( p @ ( ap @ V2Q @ V4x ) ) ) ) ) )
=> ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V1R ) @ V6a ) @ V0b ) )
=> ( p @ ( ap @ V2Q @ V6a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__INDUCT__ALT__RIGHT,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1R @ V0a ) @ V3y ) )
=> ( p @ ( ap @ V2Q @ V3y ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V2Q @ V4x ) )
& ( p @ ( ap @ ( ap @ V1R @ V4x ) @ V5y ) ) )
=> ( p @ ( ap @ V2Q @ V5y ) ) ) ) ) )
=> ! [V6b: $i] :
( ( mem @ V6b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V1R ) @ V0a ) @ V6b ) )
=> ( p @ ( ap @ V2Q @ V6b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V2x ) )
& ( p @ ( ap @ ( ap @ V1R @ V2x ) @ V3y ) ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V1R ) @ V4x ) @ V5y ) ) )
=> ( p @ ( ap @ V0P @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__lifts__transitive__relations,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27b ) @ V1Q ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V5x ) @ V6y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V2f @ V5x ) ) @ ( ap @ V2f @ V6y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__implies__one__step,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
& ( V1x != V2y ) )
=> ? [V3z: $i] :
( ( mem @ V3z @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) )
& ( V1x != V3z ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__RTC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__TC__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__RC__EQNS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__ALT__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1a ) @ V2b ) )
<=> ! [V3Q: $i] :
( ( mem @ V3Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V3Q @ V2b ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
& ( p @ ( ap @ V3Q @ V5y ) ) )
=> ( p @ ( ap @ V3Q @ V4x ) ) ) ) ) )
=> ( p @ ( ap @ V3Q @ V1a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__ALT__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( ( p @ ( ap @ V1Q @ V2b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ V1Q @ V4y ) ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V5x ) @ V2b ) )
=> ( p @ ( ap @ V1Q @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__ALT__RIGHT__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1a ) @ V2b ) )
<=> ! [V3Q: $i] :
( ( mem @ V3Q @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V3Q @ V1a ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ V3Q @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ V3Q @ V5z ) ) ) ) ) )
=> ( p @ ( ap @ V3Q @ V2b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__ALT__RIGHT__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ( p @ ( ap @ V1Q @ V2a ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( p @ ( ap @ V1Q @ V3y ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V4z ) ) )
=> ( p @ ( ap @ V1Q @ V4z ) ) ) ) ) )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2a ) @ V5z ) )
=> ( p @ ( ap @ V1Q @ V5z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V5z ) ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V6x ) @ V7y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__RULES__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V1x ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V4z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2x ) @ V4z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__STRONG__INDUCT__RIGHT1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V2x ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V3x ) @ V5z ) ) ) ) ) ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V6x ) @ V7y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6x ) @ V7y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEXTEND__RTC__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2y ) @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEXTEND__RTC__TC__EQN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2z ) )
<=> ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3y ) @ V2z ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__RC__identity,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__SC__identity,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__TC__identity,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2ERC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Erelation_2ESC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Erelation_2ETC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Erelation_2ERC__MOVES__OUT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) ) )
& ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__EQUIVALENCE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eequivalence @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__IDEM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__CASES1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( ( V1x = V2y )
| ? [V3u: $i] :
( ( mem @ V3u @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3u ) @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__CASES__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( ( V1x = V2y )
| ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__CASES2,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( ( V1x = V2y )
| ? [V3u: $i] :
( ( mem @ V3u @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V3u ) )
& ( p @ ( ap @ ( ap @ V0R @ V3u ) @ V2y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__CASES__RTC__TWICE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
<=> ? [V3u: $i] :
( ( mem @ V3u @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V3u ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3u ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__CASES1__E,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2z ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2z ) )
| ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V3y ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__CASES1,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2z ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2z ) )
| ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V3y ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__CASES2__E,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2z ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2z ) )
| ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__CASES2,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2z ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2z ) )
| ? [V3y: $i] :
( ( mem @ V3y @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V1Q ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V1Q ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__INDUCTION,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V4x ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1P @ V5x ) @ V6y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6y ) @ V5x ) ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ! [V9z: $i] :
( ( mem @ V9z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V8y ) )
& ( p @ ( ap @ ( ap @ V1P @ V8y ) @ V9z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V9z ) ) ) ) ) ) )
=> ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V10x ) @ V11y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__REFL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V1x ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__R,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__SYM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V2y ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__TRANS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V2y ) @ V3z ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V3z ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__EQC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__EQC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__EQC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__MOVES__IN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2ESTRONG__EQC__INDUCTION,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ V4x ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V5x ) @ V6y ) )
& ( p @ ( ap @ ( ap @ V1P @ V5x ) @ V6y ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V6y ) @ V5x ) ) ) ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ! [V8y: $i] :
( ( mem @ V8y @ A_27a )
=> ! [V9z: $i] :
( ( mem @ V9z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V8y ) )
& ( p @ ( ap @ ( ap @ V1P @ V8y ) @ V9z ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V7x ) @ V8y ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V8y ) @ V9z ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V7x ) @ V9z ) ) ) ) ) ) )
=> ! [V10x: $i] :
( ( mem @ V10x @ A_27a )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V10x ) @ V11y ) )
=> ( p @ ( ap @ ( ap @ V1P @ V10x ) @ V11y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EALT__equivalence,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eequivalence @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
<=> ( ( ap @ V0R @ V1x )
= ( ap @ V0R @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EEQC__MONOTONE,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R_27: $i] :
( ( mem @ V1R_27 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R_27 @ V4x ) @ V5y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V1R_27 ) @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__EQC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__lifts__monotonicities,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ ( ap @ V1f @ V4x ) ) @ ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__lifts__reflexive__transitive__relations,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) )
& ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27b ) @ V1Q ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27b ) @ V1Q ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V5x ) @ V6y ) )
=> ( p @ ( ap @ ( ap @ V1Q @ ( ap @ V2f @ V5x ) ) @ ( ap @ V2f @ V6y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__lifts__equalities,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
=> ( ( ap @ V1f @ V2x )
= ( ap @ V1f @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V4x ) @ V5y ) )
=> ( ( ap @ V1f @ V4x )
= ( ap @ V1f @ V5y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERTC__lifts__invariants,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V2x ) )
& ( p @ ( ap @ ( ap @ V1R @ V2x ) @ V3y ) ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ V0P @ V4x ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V1R ) @ V4x ) @ V5y ) ) )
=> ( p @ ( ap @ V0P @ V5y ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWF__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
<=> ! [V1B: $i] :
( ( mem @ V1B @ ( arr @ A_27a @ bool ) )
=> ( ? [V2w: $i] :
( ( mem @ V2w @ A_27a )
& ( p @ ( ap @ V1B @ V2w ) ) )
=> ? [V3min: $i] :
( ( mem @ V3min @ A_27a )
& ( p @ ( ap @ V1B @ V3min ) )
& ! [V4b: $i] :
( ( mem @ V4b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4b ) @ V3min ) )
=> ~ ( p @ ( ap @ V1B @ V4b ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__INDUCTION__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V1P @ V3y ) ) ) )
=> ( p @ ( ap @ V1P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ V1P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EINDUCTION__WF__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V1P @ V3y ) ) ) )
=> ( p @ ( ap @ V1P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ V1P @ V4x ) ) ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__EQ__INDUCTION__THM,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V1P @ V3y ) ) ) )
=> ( p @ ( ap @ V1P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ V1P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__NOT__REFL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
=> ( V1x != V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__irreflexive,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) ) ) ) ).
thf(ax_thm_2Erelation_2EEMPTY__REL__DEF,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) @ V0x ) @ V1y ) )
<=> $false ) ) ) ).
thf(conj_thm_2Erelation_2EWF__EMPTY__REL,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2EWF__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1P @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V1P ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__TC__EQN,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__noloops,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) @ V1x ) @ V2y ) )
=> ( V1x != V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__antisymmetric,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(ax_thm_2Erelation_2Einv__image__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27a @ A_27b ) @ V0R ) @ V1f )
= ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__image__thm,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27a @ A_27b ) @ V0R ) @ V1f ) @ V2x ) @ V3y ) )
<=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ V2x ) ) @ ( ap @ V1f @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27b ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27a @ A_27b ) @ V0R ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Etotal__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etotal @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a ) @ V0R ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a ) @ V0R ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a ) @ V0R ) @ V1f ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__inv__image,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2Einv__image @ A_27b @ A_27a ) @ V0R ) @ V1f ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1R ) @ V2x )
= ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ ( ap @ ( ap @ V1R @ V3y ) @ V2x ) ) @ ( ap @ V0f @ V3y ) ) @ ( c_2Ebool_2EARB @ A_27b ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERESTRICT__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1R @ V2y ) @ V3z ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V0f ) @ V1R ) @ V3z ) @ V2y )
= ( ap @ V0f @ V2y ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Eapprox__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Eapprox @ A_27a @ A_27b ) @ V0R ) @ V1M ) @ V2x ) @ V3f ) )
<=> ( V3f
= ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ V1M @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V3f ) @ V0R ) @ V4y ) ) @ V4y ) ) )
@ V0R )
@ V2x ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Ethe__fun__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Ethe__fun @ A_27a @ A_27b ) @ V0R ) @ V1M ) @ V2x )
= ( ap @ ( c_2Emin_2E_40 @ ( arr @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V3f: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Eapprox @ A_27a @ A_27b ) @ V0R ) @ V1M ) @ V2x ) @ V3f ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWFREC__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V0R ) @ V1M )
= ( lam @ A_27a
@ ^ [V2x: $i] :
( ap
@ ( ap @ V1M
@ ( ap
@ ( ap
@ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b )
@ ( ap
@ ( ap @ ( ap @ ( c_2Erelation_2Ethe__fun @ A_27a @ A_27b ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V3f: $i] :
( lam @ A_27a
@ ^ [V4v: $i] : ( ap @ ( ap @ V1M @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V3f ) @ V0R ) @ V4v ) ) @ V4v ) ) ) )
@ V2x ) )
@ V0R )
@ V2x ) )
@ V2x ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFREC__THM,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V0R ) @ V1M ) @ V2x )
= ( ap @ ( ap @ V1M @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V0R ) @ V1M ) ) @ V0R ) @ V2x ) ) @ V2x ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFREC__COROLLARY,axiom,
! [A_27a: del,A_27b: del,V0M: $i] :
( ( mem @ V0M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ( ( V2f
= ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V1R ) @ V0M ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V1R ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ V2f @ V3x )
= ( ap @ ( ap @ V0M @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V2f ) @ V1R ) @ V3x ) ) @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__RECURSION__THM,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
=> ! [V1M: $i] :
( ( mem @ V1M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( p
@ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ A_27a @ A_27b )
@ ^ [V2f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ ( ap @ V1M @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERESTRICT @ A_27a @ A_27b ) @ V2f ) @ V0R ) @ V3x ) ) @ V3x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWFP__DEF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V1a ) )
<=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V3x ) )
=> ( p @ ( ap @ V2P @ V4y ) ) ) )
=> ( p @ ( ap @ V2P @ V3x ) ) ) )
=> ( p @ ( ap @ V2P @ V1a ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFP__RULES,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V2y ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFP__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V1P @ V3y ) ) ) )
=> ( p @ ( ap @ V1P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V4x ) )
=> ( p @ ( ap @ V1P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFP__CASES,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V1x ) )
<=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V2y ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWFP__STRONG__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V1R ) @ V2x ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V1R @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) ) )
=> ( p @ ( ap @ V0P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V1R ) @ V4x ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWF__EQ__WFP,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2EWFP @ A_27a ) @ V0R ) @ V1x ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EINDUCTIVE__INVARIANT__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b ) @ V0R ) @ V1P ) @ V2M ) )
<=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V5y ) @ V4x ) )
=> ( p @ ( ap @ ( ap @ V1P @ V5y ) @ ( ap @ V3f @ V5y ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ ( ap @ ( ap @ V2M @ V3f ) @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1D: $i] :
( ( mem @ V1D @ ( arr @ A_27a @ bool ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b ) @ V0R ) @ V1D ) @ V2P ) @ V3M ) )
<=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ V1D @ V5x ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( p @ ( ap @ V1D @ V6y ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V6y ) @ V5x ) )
=> ( p @ ( ap @ ( ap @ V2P @ V6y ) @ ( ap @ V4f @ V6y ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ V2P @ V5x ) @ ( ap @ ( ap @ V3M @ V4f ) @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EINDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b ) @ V0R ) @ V1P ) @ V2M ) ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ V1P @ V3x ) @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V0R ) @ V2M ) @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__WFREC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2P: $i] :
( ( mem @ V2P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( V0f
= ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V1R ) @ V3M ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V1R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT @ A_27a @ A_27b ) @ V1R ) @ V2P ) @ V3M ) ) )
=> ( p @ ( ap @ ( ap @ V2P @ V4x ) @ ( ap @ V0f @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EINDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V3D: $i] :
( ( mem @ V3D @ ( arr @ A_27a @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b ) @ V0R ) @ V3D ) @ V1P ) @ V2M ) )
& ( p @ ( ap @ V3D @ V4x ) ) )
=> ( p @ ( ap @ ( ap @ V1P @ V4x ) @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V0R ) @ V2M ) @ V4x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ETFL__INDUCTIVE__INVARIANT__ON__WFREC,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2D: $i] :
( ( mem @ V2D @ ( arr @ A_27a @ bool ) )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V4M: $i] :
( ( mem @ V4M @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( V0f
= ( ap @ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ A_27b ) @ V1R ) @ V4M ) )
& ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V1R ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EINDUCTIVE__INVARIANT__ON @ A_27a @ A_27b ) @ V1R ) @ V2D ) @ V3P ) @ V4M ) )
& ( p @ ( ap @ V2D @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ V3P @ V5x ) @ ( ap @ V0f @ V5x ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Einv__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27b )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27b ) @ V0R ) @ V1x ) @ V2y ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V1x ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__inv,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27b @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27b ) @ V0R ) )
= V0R ) ) ).
thf(conj_thm_2Erelation_2Einv__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__SC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__TC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__EQC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__MOVES__OUT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= V0R )
& ( ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2ESC @ A_27a ) @ V0R ) )
& ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
& ( ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) )
& ( ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) )
& ( ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) )
= ( ap @ ( c_2Erelation_2EEQC @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__inv,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Eirreflexive__inv,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__inv,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Eantisymmetric__inv,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__inv,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__inv__identity,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2Eequivalence__inv__identity,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Eequivalence @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R )
= V0R ) ) ) ).
thf(ax_thm_2Erelation_2EINVOL__DEF,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EINVOL @ A_27z ) @ V0f ) )
<=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27z @ A_27z @ A_27z ) @ V0f ) @ V0f )
= ( c_2Ecombin_2EI @ A_27z ) ) ) ) ).
thf(conj_thm_2Erelation_2EINVOL,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EINVOL @ A_27z ) @ V0f ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27z )
=> ( ( ap @ V0f @ ( ap @ V0f @ V1x ) )
= V1x ) ) ) ) ).
thf(conj_thm_2Erelation_2EINVOL__ONE__ONE,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EINVOL @ A_27z ) @ V0f ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27z )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27z )
=> ( ( ( ap @ V0f @ V1a )
= ( ap @ V0f @ V2b ) )
<=> ( V1a = V2b ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EINVOL__ONE__ENO,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EINVOL @ A_27z ) @ V0f ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27z )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27z )
=> ( ( ( ap @ V0f @ V1a )
= V2b )
<=> ( V1a
= ( ap @ V0f @ V2b ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ENOT__INVOL,axiom,
p @ ( ap @ ( c_2Erelation_2EINVOL @ bool ) @ c_2Ebool_2E_7E ) ).
thf(ax_thm_2Erelation_2EIDEM__DEF,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EIDEM @ A_27z ) @ V0f ) )
<=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27z @ A_27z @ A_27z ) @ V0f ) @ V0f )
= V0f ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM,axiom,
! [A_27z: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27z @ A_27z ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EIDEM @ A_27z ) @ V0f ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27z )
=> ( ( ap @ V0f @ ( ap @ V0f @ V1x ) )
= ( ap @ V0f @ V1x ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__INVOL,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EINVOL @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) ) ) ).
thf(ax_thm_2Erelation_2EO__DEF,axiom,
! [A_27g: del,A_27h: del,A_27k: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27h @ ( arr @ A_27k @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27g @ ( arr @ A_27h @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27g )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27k )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27g @ A_27h @ A_27k ) @ V0R1 ) @ V1R2 ) @ V2x ) @ V3z ) )
<=> ? [V4y: $i] :
( ( mem @ V4y @ A_27h )
& ( p @ ( ap @ ( ap @ V1R2 @ V2x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V4y ) @ V3z ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__O,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R_27: $i] :
( ( mem @ V1R_27 @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b ) @ V0R ) @ V1R_27 ) )
= ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27b @ A_27a @ A_27c ) @ ( ap @ ( c_2Erelation_2Einv @ A_27c @ A_27a ) @ V1R_27 ) ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27b ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERSUBSET,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2x ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Eirreflexive__RSUBSET,axiom,
! [A_27a: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V1R2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ V0R1 ) @ V1R2 ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R1 ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERUNION,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2x ) @ V3y ) )
| ( p @ ( ap @ ( ap @ V1R2 @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERUNION__COMM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 )
= ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V1R2 ) @ V0R1 ) ) ) ) ).
thf(conj_thm_2Erelation_2ERUNION__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V1R2 ) @ V2R3 ) )
= ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2R3 ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERINTER,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R1 @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ V1R2 @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERINTER__COMM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 )
= ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V1R2 ) @ V0R1 ) ) ) ) ).
thf(conj_thm_2Erelation_2ERINTER__ASSOC,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V1R2 ) @ V2R3 ) )
= ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2R3 ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Eantisymmetric__RINTER,axiom,
! [A_27a: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R1 ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a ) @ V0R1 ) @ V1R2 ) ) ) )
& ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V1R2 ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__RINTER,axiom,
! [A_27a: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R1 ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V1R2 ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a ) @ V0R1 ) @ V1R2 ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERCOMPL,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERCOMPL @ A_27a @ A_27b ) @ V0R ) @ V1x ) @ V2y ) )
<=> ~ ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ereflexive__Id__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Esymmetric__inv__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Esymmetric @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Etransitive__O__RSUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27a ) @ V0R ) @ V0R ) ) @ V0R ) ) ) ) ).
thf(ax_thm_2Erelation_2EPreOrder,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EPreOrder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EOrder,axiom,
! [A_27g: del,V0Z: $i] :
( ( mem @ V0Z @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27g ) @ V0Z ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27g ) @ V0Z ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27g ) @ V0Z ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWeakOrder,axiom,
! [A_27g: del,V0Z: $i] :
( ( mem @ V0Z @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27g ) @ V0Z ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Ereflexive @ A_27g ) @ V0Z ) )
& ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27g ) @ V0Z ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27g ) @ V0Z ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EStrongOrder,axiom,
! [A_27g: del,V0Z: $i] :
( ( mem @ V0Z @ ( arr @ A_27g @ ( arr @ A_27g @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27g ) @ V0Z ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27g ) @ V0Z ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27g ) @ V0Z ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Eirrefl__trans__implies__antisym,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Eirreflexive @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etransitive @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EStrongOrd__Ord,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EWeakOrd__Ord,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2EWeakOrder__EQ,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ V0R ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ( ( V1y = V2z )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1y ) @ V2z ) )
& ( p @ ( ap @ ( ap @ V0R @ V2z ) @ V1y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERSUBSET__ANTISYM,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V1R2 ) @ V0R1 ) ) )
=> ( V0R1 = V1R2 ) ) ) ) ).
thf(conj_thm_2Erelation_2ERSUBSET__antisymmetric,axiom,
! [A_27a: del,A_27b: del] : ( p @ ( ap @ ( c_2Erelation_2Eantisymmetric @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Erelation_2ERSUBSET__WeakOrder,axiom,
! [A_27a: del,A_27b: del] : ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Erelation_2EEqIsBothRSUBSET,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1z: $i] :
( ( mem @ V1z @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( V0y = V1z )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V0y ) @ V1z ) )
& ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V1z ) @ V0y ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ESTRORD,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R ) @ V1a ) @ V2b ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b ) )
& ( V1a != V2b ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESTRORD__AND__NOT__Id,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R )
= ( ap @ ( ap @ ( c_2Erelation_2ERINTER @ A_27a @ A_27a ) @ V0R ) @ ( ap @ ( c_2Erelation_2ERCOMPL @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__OR__Id,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R )
= ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27a ) @ V0R ) @ ( c_2Emin_2E_3D @ A_27a ) ) ) ) ).
thf(conj_thm_2Erelation_2ERC__Weak,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESTRORD__Strong,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ESTRORD__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2ERC__STRORD,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ V0R ) )
=> ( ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R ) )
= V0R ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM__STRORD,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EIDEM @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2ESTRORD @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM__RC,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EIDEM @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2ERC @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM__SC,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EIDEM @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2ESC @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM__TC,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EIDEM @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2ETC @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2EIDEM__RTC,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EIDEM @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) @ ( c_2Erelation_2ERTC @ A_27a ) ) ) ).
thf(conj_thm_2Erelation_2Etrichotomous__STRORD,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ ( ap @ ( c_2Erelation_2ESTRORD @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) ) ) ) ).
thf(conj_thm_2Erelation_2Etrichotomous__RC,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) ) ) ) ).
thf(ax_thm_2Erelation_2ELinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2ELinearOrder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2EOrder @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EStrongLinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongLinearOrder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2EStrongOrder @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWeakLinearOrder,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2Etrichotomous @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EWeakLinearOrder__dichotomy,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakLinearOrder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Erelation_2EWeakOrder @ A_27a ) @ V0R ) )
& ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1a ) @ V2b ) )
| ( p @ ( ap @ ( ap @ V0R @ V2b ) @ V1a ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Ediag__def,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2Ediag @ A_27a ) @ V0A ) @ V1x ) @ V2y ) )
<=> ( ( V1x = V2y )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V0A ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EO__ASSOC,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27c @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27d @ ( arr @ A_27c @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27a @ ( arr @ A_27d @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27c @ A_27b ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27d @ A_27c ) @ V1R2 ) @ V2R3 ) )
= ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27d @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27d @ A_27c @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2R3 ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EId__O,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27b @ A_27b ) @ ( c_2Emin_2E_3D @ A_27b ) ) @ V0R )
= V0R ) ) ).
thf(conj_thm_2Erelation_2EO__Id,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27a @ A_27a @ A_27b ) @ V0R ) @ ( c_2Emin_2E_3D @ A_27a ) )
= V0R ) ) ).
thf(conj_thm_2Erelation_2EO__MONO,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2S1: $i] :
( ( mem @ V2S1 @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ! [V3S2: $i] :
( ( mem @ V3S2 @ ( arr @ A_27c @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27c @ A_27a ) @ V2S1 ) @ V3S2 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b ) @ V0R1 ) @ V2S1 ) ) @ ( ap @ ( ap @ ( c_2Erelation_2EO @ A_27c @ A_27a @ A_27b ) @ V1R2 ) @ V3S2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Einv__Id,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) )
= ( c_2Emin_2E_3D @ A_27a ) ) ).
thf(conj_thm_2Erelation_2Einv__diag,axiom,
! [A_27a: del,V0A: $i] :
( ( mem @ V0A @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ ( ap @ ( c_2Erelation_2Ediag @ A_27a ) @ V0A ) )
= ( ap @ ( c_2Erelation_2Ediag @ A_27a ) @ V0A ) ) ) ).
thf(ax_thm_2Erelation_2ERDOM__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V0R ) @ V1x ) )
<=> ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EIN__RDOM,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V1R ) ) )
<=> ? [V2y: $i] :
( ( mem @ V2y @ A_27b )
& ( p @ ( ap @ ( ap @ V1R @ V0x ) @ V2y ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERRANGE,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERRANGE @ A_27a @ A_27b ) @ V0R ) @ V1y ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V1y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EIN__RRANGE,axiom,
! [A_27a: del,A_27b: del,V0y: $i] :
( ( mem @ V0y @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27b @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0y ) @ ( ap @ ( c_2Erelation_2ERRANGE @ A_27b @ A_27a ) @ V1R ) ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( p @ ( ap @ ( ap @ V1R @ V2x ) @ V0y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EIN__RDOM__RUNION,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1R1: $i] :
( ( mem @ V1R1 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2R2: $i] :
( ( mem @ V2R2 @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2ERUNION @ A_27a @ A_27b ) @ V1R1 ) @ V2R2 ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V1R1 ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V2R2 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERUNIV,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) @ V0x ) @ V1y ) )
<=> $true ) ) ) ).
thf(conj_thm_2Erelation_2ERUNIV__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) ) @ V0R ) )
<=> ( V0R
= ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27b ) @ V0R ) @ ( c_2Erelation_2ERUNIV @ A_27a @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EREMPTY__SUBSET,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) @ V0R ) )
& ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2ERSUBSET @ A_27a @ A_27a ) @ V0R ) @ ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) )
<=> ( V0R
= ( c_2Erelation_2EEMPTY__REL @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERRESTRICT__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERRESTRICT @ A_27a @ A_27b ) @ V0R ) @ V1s ) @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EIN__RDOM__RRESTRICT,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2ERRESTRICT @ A_27a @ A_27b ) @ V1R ) @ V2s ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V1R ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ V2s ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ERDOM__DELETE__DEF,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2u: $i] :
( ( mem @ V2u @ A_27a )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERDOM__DELETE @ A_27a @ A_27b ) @ V0R ) @ V1x ) @ V2u ) @ V3v ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V2u ) @ V3v ) )
& ( V2u != V1x ) ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2EIN__RDOM__DELETE,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V2k: $i] :
( ( mem @ V2k @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Erelation_2ERDOM__DELETE @ A_27a @ A_27b ) @ V1R ) @ V2k ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Erelation_2ERDOM @ A_27a @ A_27b ) @ V1R ) ) )
& ( V0x != V2k ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Ediamond__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) )
=> ? [V4u: $i] :
( ( mem @ V4u @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V2y ) @ V4u ) )
& ( p @ ( ap @ ( ap @ V0R @ V3z ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Ercdiamond__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ercdiamond @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) )
=> ? [V4u: $i] :
( ( mem @ V4u @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V2y ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) @ V3z ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ECR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2ECR @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2EWCR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWCR @ A_27a ) @ V0R ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V3z ) ) )
=> ? [V4u: $i] :
( ( mem @ V4u @ A_27a )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V2y ) @ V4u ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Erelation_2ERTC @ A_27a ) @ V0R ) @ V3z ) @ V4u ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Erelation_2ESN__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2ESN @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Erelation_2Einv @ A_27a @ A_27a ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Erelation_2Enf__def,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Erelation_2Enf @ A_27a @ A_27b ) @ V0R ) @ V1x ) )
<=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ~ ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ercdiamond__diamond,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ercdiamond @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ediamond__RC__diamond,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ ( ap @ ( c_2Erelation_2ERC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Ediamond__TC__diamond,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ ( ap @ ( c_2Erelation_2ETC @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2Eestablish__CR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2Ercdiamond @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2ECR @ A_27a ) @ V0R ) ) )
& ( ( p @ ( ap @ ( c_2Erelation_2Ediamond @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Erelation_2ECR @ A_27a ) @ V0R ) ) ) ) ) ).
thf(conj_thm_2Erelation_2ENewmans__lemma,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( ( p @ ( ap @ ( c_2Erelation_2EWCR @ A_27a ) @ V0R ) )
& ( p @ ( ap @ ( c_2Erelation_2ESN @ A_27a ) @ V0R ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2ECR @ A_27a ) @ V0R ) ) ) ) ).
%------------------------------------------------------------------------------