ITP001 Axioms: ITP057^5.ax
%------------------------------------------------------------------------------
% File : ITP057^5 : 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 : wellorder^2.ax [Gau20]
% : HL4057^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 170 ( 6 unt; 23 typ; 0 def)
% Number of atoms : 3563 ( 94 equ; 0 cnn)
% Maximal formula atoms : 80 ( 20 avg)
% Number of connectives : 6835 ( 11 ~; 13 |; 125 &;6158 @)
% ( 43 <=>; 485 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 16 avg;6158 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 25 ( 25 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 80 usr; 59 con; 0-2 aty)
% Number of variables : 629 ( 32 ^ 564 !; 33 ?; 629 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Ewellorder_2Ewellorder,type,
ty_2Ewellorder_2Ewellorder: del > del ).
thf(tp_c_2Ewellorder_2EADD1,type,
c_2Ewellorder_2EADD1: del > $i ).
thf(mem_c_2Ewellorder_2EADD1,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2EADD1 @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) ) ) ).
thf(tp_c_2Ewellorder_2EChain,type,
c_2Ewellorder_2EChain: del > $i ).
thf(mem_c_2Ewellorder_2EChain,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2EChain @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2EelsOf,type,
c_2Ewellorder_2EelsOf: del > $i ).
thf(mem_c_2Ewellorder_2EelsOf,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2Efinite,type,
c_2Ewellorder_2Efinite: del > $i ).
thf(mem_c_2Ewellorder_2Efinite,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Efinite @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ bool ) ) ).
thf(tp_c_2Ewellorder_2Efl,type,
c_2Ewellorder_2Efl: del > $i ).
thf(mem_c_2Ewellorder_2Efl,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Efl @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2EfromNatWO,type,
c_2Ewellorder_2EfromNatWO: del > $i ).
thf(mem_c_2Ewellorder_2EfromNatWO,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2EfromNatWO @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( ty_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).
thf(tp_c_2Ewellorder_2Eiseg,type,
c_2Ewellorder_2Eiseg: del > $i ).
thf(mem_c_2Ewellorder_2Eiseg,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Ewellorder_2Eorderiso,type,
c_2Ewellorder_2Eorderiso: del > del > $i ).
thf(mem_c_2Ewellorder_2Eorderiso,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2Eorderlt,type,
c_2Ewellorder_2Eorderlt: del > del > $i ).
thf(mem_c_2Ewellorder_2Eorderlt,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2Eposet,type,
c_2Ewellorder_2Eposet: del > $i ).
thf(mem_c_2Ewellorder_2Eposet,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Eposet @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Ewellorder_2Eremove,type,
c_2Ewellorder_2Eremove: del > $i ).
thf(mem_c_2Ewellorder_2Eremove,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Eremove @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) ) ) ).
thf(tp_c_2Ewellorder_2EwZERO,type,
c_2Ewellorder_2EwZERO: del > $i ).
thf(mem_c_2Ewellorder_2EwZERO,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2EwZERO @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) ).
thf(tp_c_2Ewellorder_2Ewellfounded,type,
c_2Ewellorder_2Ewellfounded: del > $i ).
thf(mem_c_2Ewellorder_2Ewellfounded,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Ewellorder_2Ewellorder,type,
c_2Ewellorder_2Ewellorder: del > $i ).
thf(mem_c_2Ewellorder_2Ewellorder,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ bool ) ) ).
thf(tp_c_2Ewellorder_2Ewellorder__ABS,type,
c_2Ewellorder_2Ewellorder__ABS: del > $i ).
thf(mem_c_2Ewellorder_2Ewellorder__ABS,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ ( arr @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) ) ).
thf(tp_c_2Ewellorder_2Ewellorder__REP,type,
c_2Ewellorder_2Ewellorder__REP: del > $i ).
thf(mem_c_2Ewellorder_2Ewellorder__REP,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Ewellorder_2Ewleast,type,
c_2Ewellorder_2Ewleast: del > $i ).
thf(mem_c_2Ewellorder_2Ewleast,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( ty_2Eoption_2Eoption @ A_27a ) ) ) ) ).
thf(tp_c_2Ewellorder_2Ewo2wo,type,
c_2Ewellorder_2Ewo2wo: del > del > $i ).
thf(mem_c_2Ewellorder_2Ewo2wo,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) ) ) ) ).
thf(tp_c_2Ewellorder_2Ewobound,type,
c_2Ewellorder_2Ewobound: del > $i ).
thf(mem_c_2Ewellorder_2Ewobound,axiom,
! [A_27a: del] : ( mem @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewellfounded__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ V0R ) )
<=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ? [V2w: $i] :
( ( mem @ V2w @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2w ) @ V1s ) ) )
=> ? [V3min: $i] :
( ( mem @ V3min @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3min ) @ V1s ) )
& ! [V4w: $i] :
( ( mem @ V4w @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4w ) @ V3min ) ) @ V0R ) )
=> ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4w ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellfounded__WF,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ V0R ) )
<=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Epair_2ECURRY @ A_27a @ A_27a @ bool ) @ V0R ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewellorder__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ V0R ) )
<=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0R ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0R ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0R ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0R ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__EMPTY,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__SING,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) )
<=> ( V0x = V1y ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Errestrict__SUBSET,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) @ V0r ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellfounded__subset,axiom,
! [A_27a: del,V0r0: $i] :
( ( mem @ V0r0 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ V1r ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r0 ) @ V1r ) ) )
=> ( p @ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a ) @ V0r0 ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewellorder__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) @ ( c_2Ewellorder_2Ewellorder @ A_27a ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewellorder__ABSREP,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ V1r ) )
<=> ( ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EmkWO__destWO,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0a ) )
= V0a ) ) ).
thf(conj_thm_2Ewellorder_2EdestWO__mkWO,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ V0r ) )
=> ( ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ V0r ) )
= V0r ) ) ) ).
thf(ax_thm_2Ewellorder_2EelsOf__def,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__elsOf,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V2w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__elsOf,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V2w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__trichotomy,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) )
| ( V1x = V2y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__REFL,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V0x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) )
<=> $false ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1y ) @ V3z ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V3z ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__ANTISYM,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1y ) @ V0x ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
=> ( V0x = V1y ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__WLE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__WLE,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V0x ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Etransitive__strict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__TRANS,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1y ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V3z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__WF,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( p
@ ( ap @ ( c_2Ewellorder_2Ewellfounded @ A_27a )
@ ( lam @ ( ty_2Epair_2Eprod @ A_27a @ A_27a )
@ ^ [V1p: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V1p ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__WF2,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( p
@ ( ap @ ( c_2Erelation_2EWF @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Eiseg__def,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V0w ) @ V1x )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2y ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Estrict__subset,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r1 ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V1r2 ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Etransitive__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eantisym__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Elinear__order__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ereflexive__rrestrict,axiom,
! [A_27a: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Ereflexive @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ V0r ) @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__rrestrict,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) @ V1s ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewobound__def,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0x ) @ V1w )
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ ( ap @ ( ap @ ( c_2Eset__relation_2Errestrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V1w ) @ V0x ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__wobound,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ! [V3w: $i] :
( ( mem @ V3w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V2z ) @ V3w ) ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V2z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1y ) @ V2z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__wobound,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2z: $i] :
( ( mem @ V2z @ A_27a )
=> ! [V3w: $i] :
( ( mem @ V3w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V2z ) @ V3w ) ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V2z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1y ) @ V2z ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__cases,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
& ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ V1s ) )
& ( V0w
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ V1s ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWEXTENSION,axiom,
! [A_27a: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( V0w1 = V1w2 )
<=> ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2a ) @ V3b ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2a ) @ V3b ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewobound2,axiom,
! [A_27a: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0a ) @ V1b ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V1b ) @ V2w ) )
= ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0a ) @ V2w ) ) ) ) ) ) ).
thf(stp_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,type,
tp__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o: $tType ).
thf(stp_inj_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,type,
inj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o: tp__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o > $i ).
thf(stp_surj_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,type,
surj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o: $i > tp__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o ).
thf(stp_inj_surj_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,axiom,
! [X: tp__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o] :
( ( surj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o @ ( inj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o @ X ) )
= X ) ).
thf(stp_inj_mem_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,axiom,
! [X: tp__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o] : ( mem @ ( inj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o @ X ) @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ bool ) ) ).
thf(stp_iso_mem_c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o,axiom,
! [X: $i] :
( ( mem @ X @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ bool ) )
=> ( X
= ( inj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o @ ( surj__c_ty_2Epair_2Eprod_c_ty_2Epair_2Eprod_ty_2Enum_2Enum_ty_2Enum_2Enum_o @ X ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__fromNat,axiom,
! [V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( c_2Ewellorder_2Ewellorder @ ty_2Enum_2Enum )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V2j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) @ V1i ) @ V2j ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1i ) @ V2j ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2j ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EINJ__preserves__transitive,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Etransitive @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27a @ A_27b @ A_27b ) @ V1f ) @ V1f ) ) @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EINJ__preserves__antisym,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Eset__relation_2Eantisym @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27a @ A_27b @ A_27b ) @ V1f ) @ V1f ) ) @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EINJ__preserves__linear__order,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27a ) @ V0r ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Eset__relation_2Elinear__order @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27a @ A_27b @ A_27b ) @ V1f ) @ V1f ) ) @ V0r ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Edomain__IMAGE__ff,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27a ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27d @ A_27b ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27a @ A_27b ) @ V0f ) @ V1g ) ) @ V2r ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27c @ A_27a ) @ V0f ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27c @ A_27d ) @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Erange__IMAGE__ff,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27c @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27d @ A_27a ) )
=> ! [V2r: $i] :
( ( mem @ V2r @ ( arr @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27c @ A_27d ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27c @ A_27d @ A_27b @ A_27a ) @ V0f ) @ V1g ) ) @ V2r ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27d @ A_27a ) @ V1g ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27d @ A_27c ) @ V2r ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EINJ__preserves__wellorder,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a ) @ V0r ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( c_2Eset__relation_2Edomain @ A_27a @ A_27a ) @ V0r ) ) @ ( ap @ ( c_2Eset__relation_2Erange @ A_27a @ A_27a ) @ V0r ) ) ) @ V2t ) ) )
=> ( p @ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27a @ A_27a @ A_27b @ A_27b ) @ V1f ) @ V1f ) ) @ V0r ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__fromNat__SUM,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( p
@ ( ap @ ( c_2Ewellorder_2Ewellorder @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V2j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Esum_2EINL @ ty_2Enum_2Enum @ A_27a ) @ V1i ) ) @ ( ap @ ( c_2Esum_2EINL @ ty_2Enum_2Enum @ A_27a ) @ V2j ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1i ) @ V2j ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2j ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2EfromNatWO__def,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Ewellorder_2EfromNatWO @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ty_2Epair_2Eprod @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ty_2Enum_2Enum @ ty_2Enum_2Enum @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] :
( lam @ ty_2Enum_2Enum
@ ^ [V2j: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Esum_2EINL @ ty_2Enum_2Enum @ A_27a ) @ V1i ) ) @ ( ap @ ( c_2Esum_2EINL @ ty_2Enum_2Enum @ A_27a ) @ V2j ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ V1i ) @ V2j ) ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2j ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EfromNatWO__11,axiom,
! [A_27a: del,V0i: tp__ty_2Enum_2Enum,V1j: tp__ty_2Enum_2Enum] :
( ( ( ap @ ( c_2Ewellorder_2EfromNatWO @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) )
= ( ap @ ( c_2Ewellorder_2EfromNatWO @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1j ) ) )
<=> ( V0i = V1j ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__fromNatWO,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum] :
( ( ap @ ( c_2Ewellorder_2EelsOf @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ ( c_2Ewellorder_2EfromNatWO @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ty_2Enum_2Enum @ ( ty_2Esum_2Esum @ ty_2Enum_2Enum @ A_27a ) ) @ ( c_2Esum_2EINL @ ty_2Enum_2Enum @ A_27a ) ) @ ( ap @ c_2Epred__set_2Ecount @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__WIN,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) )
=> ( ( V0x = V1y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__wobound,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V0x ) @ V1w ) )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V2y ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V0x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Eorderiso__def,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w1 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) ) ) )
& ! [V4x1: $i] :
( ( mem @ V4x1 @ A_27a )
=> ! [V5x2: $i] :
( ( mem @ V5x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x1 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x2 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w1 ) ) ) )
=> ( ( ( ap @ V2f @ V4x1 )
= ( ap @ V2f @ V5x2 ) )
<=> ( V4x1 = V5x2 ) ) ) ) )
& ! [V6y: $i] :
( ( mem @ V6y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V6y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) )
=> ? [V7x: $i] :
( ( mem @ V7x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w1 ) ) )
& ( ( ap @ V2f @ V7x )
= V6y ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V8x ) @ V9y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V2f @ V8x ) ) @ ( ap @ V2f @ V9y ) ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V1w2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__thm,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
<=> ? [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V2f ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w1 ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3x ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V1w2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__REFL,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27a ) @ V0w ) @ V0w ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__SYM,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27b @ A_27a ) @ V1w2 ) @ V0w1 ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2w3: $i] :
( ( mem @ V2w3 @ ( ty_2Ewellorder_2Ewellorder @ A_27c ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27b @ A_27c ) @ V1w2 ) @ V2w3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27c ) @ V0w1 ) @ V2w3 ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Eorderlt__def,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
<=> ? [V2x: $i] :
( ( mem @ V2x @ A_27b )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27b ) @ V2x ) @ V1w2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderlt__REFL,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27a ) @ V0w ) @ V0w ) )
<=> $false ) ) ).
thf(conj_thm_2Ewellorder_2Ewobounds__preserve__bijections,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1w1: $i] :
( ( mem @ V1w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2w2: $i] :
( ( mem @ V2w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V2w2 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V0f @ V4x ) ) @ ( ap @ V0f @ V5y ) ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V2w2 ) ) ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27a ) @ V3x ) @ V1w1 ) ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewobound @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2w2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderlt__TRANS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2w3: $i] :
( ( mem @ V2w3 @ ( ty_2Ewellorder_2Ewellorder @ A_27c ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27b @ A_27c ) @ V1w2 ) @ V2w3 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27c ) @ V0w1 ) @ V2w3 ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewleast__def,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ V0w ) @ V1s )
= ( ap @ ( c_2Eoption_2Esome @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) )
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) ) )
@ ( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V3y ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Ewo2wo__def,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 )
= ( ap
@ ( ap @ ( c_2Erelation_2EWFREC @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) ) ) ) )
@ ( lam @ ( arr @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ^ [V4f: $i] :
( lam @ A_27a
@ ^ [V5x: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ bool ) @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ( lam @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ bool )
@ ^ [V6s0: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ A_27b @ bool ) @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V7s1: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ bool ) ) @ V7s1 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27b ) @ V1w2 ) @ V7s1 ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27b ) @ A_27b ) @ ( c_2Eoption_2ETHE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ V6s0 ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ V4f ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V0w1 ) @ V5x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__thm,axiom,
! [A_27a: del,A_27b: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w ) @ V1w2 ) @ V2x )
= ( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ bool ) @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ( lam @ ( arr @ ( ty_2Eoption_2Eoption @ A_27b ) @ bool )
@ ^ [V3s0: $i] :
( ap
@ ( ap @ ( c_2Ebool_2ELET @ ( arr @ A_27b @ bool ) @ ( ty_2Eoption_2Eoption @ A_27b ) )
@ ( lam @ ( arr @ A_27b @ bool )
@ ^ [V4s1: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( arr @ A_27b @ bool ) ) @ V4s1 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27b ) @ V1w2 ) @ V4s1 ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27b ) @ A_27b ) @ ( c_2Eoption_2ETHE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ V3s0 ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w ) @ V1w2 ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V0w ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewleast__IN__wo,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ V0w ) @ V1s )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
& ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) )
& ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ( V2x != V3y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewleast__EQ__NONE,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ V0w ) @ V1s )
= ( c_2Eoption_2ENONE @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__IN__w2,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) @ V2x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3y ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1w2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EIMAGE__wo2wo__SUBSET,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27a ) @ A_27a ) @ ( c_2Eoption_2ETHE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27b @ A_27a ) @ V0w1 ) @ V1w2 ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27b ) @ V0w1 ) @ V2x ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w2 ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__EQ__NONE,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) @ V2x )
= ( c_2Eoption_2ENONE @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) @ V3y )
= ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__EQ__SOME__downwards,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) @ V2x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3y ) )
=> ! [V4x0: $i] :
( ( mem @ V4x0 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x0 ) @ V2x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w1 ) ) ) )
=> ? [V5y0: $i] :
( ( mem @ V5y0 @ A_27b )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) @ V4x0 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V5y0 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Emono__woseg,axiom,
! [A_27a: del,A_27b: del,V0x1: $i] :
( ( mem @ V0x1 @ A_27a )
=> ! [V1x2: $i] :
( ( mem @ V1x2 @ A_27a )
=> ! [V2w1: $i] :
( ( mem @ V2w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V3w2: $i] :
( ( mem @ V3w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x1 ) @ V1x2 ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27b ) @ A_27b ) @ ( c_2Eoption_2ETHE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V2w1 ) @ V3w2 ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V2w1 ) @ V0x1 ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27b ) @ A_27b ) @ ( c_2Eoption_2ETHE @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ ( ty_2Eoption_2Eoption @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V2w1 ) @ V3w2 ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27a ) @ V2w1 ) @ V1x2 ) ) ) @ ( c_2Eoption_2ENONE @ A_27b ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__11,axiom,
! [A_27a: del,A_27b: del,V0x1: $i] :
( ( mem @ V0x1 @ A_27a )
=> ! [V1w1: $i] :
( ( mem @ V1w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2x2: $i] :
( ( mem @ V2x2 @ A_27a )
=> ! [V3w2: $i] :
( ( mem @ V3w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x1 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x2 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V1w1 ) @ V3w2 ) @ V0x1 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V4y ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V1w1 ) @ V3w2 ) @ V2x2 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V4y ) ) )
=> ( V0x1 = V2x2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewleast__SUBSET,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3s2: $i] :
( ( mem @ V3s2 @ ( arr @ A_27a @ bool ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ V0w ) @ V1s1 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V2x ) )
& ( ( ap @ ( ap @ ( c_2Ewellorder_2Ewleast @ A_27a ) @ V0w ) @ V3s2 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V4y ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s1 ) @ V3s2 ) ) )
=> ( ( V2x = V4y )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V4y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V0w ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__mono,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2x0: $i] :
( ( mem @ V2x0 @ A_27b )
=> ! [V3y0: $i] :
( ( mem @ V3y0 @ A_27a )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27b @ A_27a ) @ V0w1 ) @ V1w2 ) @ V2x0 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V3y0 ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27b @ A_27a ) @ V0w1 ) @ V1w2 ) @ V4x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27a ) @ V5y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ V2x0 ) @ V4x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V0w1 ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y0 ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__ONTO,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1w1: $i] :
( ( mem @ V1w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2w2: $i] :
( ( mem @ V2w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27b )
=> ! [V4y0: $i] :
( ( mem @ V4y0 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V1w1 ) @ V2w2 ) @ V0x )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V3y ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ V4y0 ) @ V3y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V2w2 ) ) ) ) )
=> ? [V5x0: $i] :
( ( mem @ V5x0 @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x0 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27a @ A_27b ) @ V1w1 ) @ V2w2 ) @ V5x0 )
= ( ap @ ( c_2Eoption_2ESOME @ A_27b ) @ V4y0 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo2wo__EQ__NONE__woseg,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27b @ A_27a ) @ V0w1 ) @ V1w2 ) @ V2x )
= ( c_2Eoption_2ENONE @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w2 )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ ( ty_2Eoption_2Eoption @ A_27a ) @ A_27a ) @ ( c_2Eoption_2ETHE @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27b @ ( ty_2Eoption_2Eoption @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Ewo2wo @ A_27b @ A_27a ) @ V0w1 ) @ V1w2 ) ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eiseg @ A_27b ) @ V0w1 ) @ V2x ) ) ) @ ( c_2Eoption_2ENONE @ A_27a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderlt__trichotomy,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
| ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27b @ A_27a ) @ V1w2 ) @ V0w1 ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2EwZERO__def,axiom,
! [A_27a: del] :
( ( c_2Ewellorder_2EwZERO @ A_27a )
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__wZERO,axiom,
! [A_27a: del] :
( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( c_2Ewellorder_2EwZERO @ A_27a ) )
= ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__wZERO,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( c_2Ewellorder_2EwZERO @ A_27a ) ) ) ) )
<=> $false ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__wZERO,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( c_2Ewellorder_2EwZERO @ A_27a ) ) ) )
<=> $false ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__wZERO,axiom,
! [A_27a: del,A_27b: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ ( c_2Ewellorder_2EwZERO @ A_27a ) ) @ V0w ) )
<=> ( V0w
= ( c_2Ewellorder_2EwZERO @ A_27b ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__EQ__EMPTY,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w )
= ( c_2Epred__set_2EEMPTY @ A_27a ) )
<=> ( V0w
= ( c_2Ewellorder_2EwZERO @ A_27a ) ) ) ) ).
thf(conj_thm_2Ewellorder_2ELT__wZERO,axiom,
! [A_27a: del,A_27b: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27b ) @ V0w ) @ ( c_2Ewellorder_2EwZERO @ A_27b ) ) )
<=> $false ) ) ).
thf(conj_thm_2Ewellorder_2Eorderlt__WF,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Erelation_2EWF @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27a ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderlt__orderiso,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0x0: $i] :
( ( mem @ V0x0 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1y0: $i] :
( ( mem @ V1y0 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V2a0: $i] :
( ( mem @ V2a0 @ ( ty_2Ewellorder_2Ewellorder @ A_27c ) )
=> ! [V3b0: $i] :
( ( mem @ V3b0 @ ( ty_2Ewellorder_2Ewellorder @ A_27d ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0x0 ) @ V1y0 ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27c @ A_27d ) @ V2a0 ) @ V3b0 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27a @ A_27c ) @ V0x0 ) @ V2a0 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderlt @ A_27b @ A_27d ) @ V1y0 ) @ V3b0 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Efinite__def,axiom,
! [A_27a: del,V0w: $i] :
( ( mem @ V0w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Efinite @ A_27a ) @ V0w ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V0w ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Efinite__iso,axiom,
! [A_27a: del,A_27b: del,V0w1: $i] :
( ( mem @ V0w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V1w2: $i] :
( ( mem @ V1w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27a @ A_27b ) @ V0w1 ) @ V1w2 ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Efinite @ A_27a ) @ V0w1 ) )
<=> ( p @ ( ap @ ( c_2Ewellorder_2Efinite @ A_27b ) @ V1w2 ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Efinite__wZERO,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Ewellorder_2Efinite @ A_27a ) @ ( c_2Ewellorder_2EwZERO @ A_27a ) ) ) ).
thf(conj_thm_2Ewellorder_2Eorderiso__unique,axiom,
! [A_27a: del,A_27b: del,V0f1: $i] :
( ( mem @ V0f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1w1: $i] :
( ( mem @ V1w1 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ! [V2w2: $i] :
( ( mem @ V2w2 @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V0f1 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V2w2 ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EBIJ @ A_27a @ A_27b ) @ V3f2 ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V2w2 ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V4x ) @ V5y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V0f1 @ V4x ) ) @ ( ap @ V0f1 @ V5y ) ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V2w2 ) ) ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6x ) @ V7y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w1 ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27b @ A_27b ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ A_27b ) @ ( ap @ V3f2 @ V6x ) ) @ ( ap @ V3f2 @ V7y ) ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27b ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27b ) @ V2w2 ) ) ) ) ) ) ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w1 ) ) )
=> ( ( ap @ V0f1 @ V8x )
= ( ap @ V3f2 @ V8x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eseteq__wlog,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ( ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ V2a ) @ V3b ) )
=> ( p @ ( ap @ ( ap @ V0P @ V3b ) @ V2a ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ! [V6b: $i] :
( ( mem @ V6b @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0P @ V5a ) @ V6b ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ ( ap @ V1f @ V5a ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V4x ) @ ( ap @ V1f @ V6b ) ) ) ) ) ) ) )
=> ! [V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ! [V8b: $i] :
( ( mem @ V8b @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0P @ V7a ) @ V8b ) )
=> ( ( ap @ V1f @ V7a )
= ( ap @ V1f @ V8b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewo__INDUCTION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) )
=> ( p @ ( ap @ V0P @ V3y ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) )
=> ( p @ ( ap @ V0P @ V2x ) ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EFORALL__NUM,axiom,
! [V0P: $i] :
( ( mem @ V0P @ ( arr @ ty_2Enum_2Enum @ bool ) )
=> ( ! [V1n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ V0P @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ V0P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Estrict__UNION,axiom,
! [A_27a: del,V0r1: $i] :
( ( mem @ V0r1 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1r2: $i] :
( ( mem @ V1r2 @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ V0r1 ) @ V1r2 ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V0r1 ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ V1r2 ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWLE__WIN__EQ,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2w: $i] :
( ( mem @ V2w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) )
<=> ( ( ( V0x = V1y )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V2w ) ) ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V2w ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Eremove__def,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2Eremove @ A_27a ) @ V0e ) @ V1w )
= ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V0e ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3y ) @ V0e ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__remove,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( p
@ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ A_27a @ A_27a @ ( ty_2Epair_2Eprod @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ V0e ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3y ) @ V0e ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__remove,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eremove @ A_27a ) @ V0e ) @ V1w ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27a ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) @ V0e ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__remove,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3w: $i] :
( ( mem @ V3w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2Eremove @ A_27a ) @ V2e ) @ V3w ) ) ) ) )
<=> ( ( V0x != V2e )
& ( V1y != V2e )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2EADD1__def,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2Ewellorder_2EADD1 @ A_27a ) @ V0e ) @ V1w )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) ) @ V1w )
@ ( ap @ ( c_2Ewellorder_2Ewellorder__ABS @ A_27a )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V0e ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0e ) @ V0e ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Ewellorder__ADD1,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0e ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) )
=> ( p
@ ( ap @ ( c_2Ewellorder_2Ewellorder @ A_27a )
@ ( ap
@ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V1w ) )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V0e ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0e ) @ V0e ) ) @ ( c_2Epred__set_2EEMPTY @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__ADD1,axiom,
! [A_27a: del,V0e: $i] :
( ( mem @ V0e @ A_27a )
=> ! [V1w: $i] :
( ( mem @ V1w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2EADD1 @ A_27a ) @ V0e ) @ V1w ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0e ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1w ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWIN__ADD1,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2e: $i] :
( ( mem @ V2e @ A_27a )
=> ! [V3w: $i] :
( ( mem @ V3w @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ ( ap @ ( ap @ ( c_2Ewellorder_2EADD1 @ A_27a ) @ V2e ) @ V3w ) ) ) ) )
<=> ( ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2e ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V3w ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V3w ) ) )
& ( V1y = V2e ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V0x ) @ V1y ) ) @ ( ap @ ( c_2Eset__relation_2Estrict @ A_27a ) @ ( ap @ ( c_2Ewellorder_2Ewellorder__REP @ A_27a ) @ V3w ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EelsOf__cardeq__iso,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ A_27a ) )
=> ! [V1wo: $i] :
( ( mem @ V1wo @ ( ty_2Ewellorder_2Ewellorder @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Epred__set_2EINJ @ A_27b @ A_27a ) @ V0f ) @ ( ap @ ( c_2Ewellorder_2EelsOf @ A_27b ) @ V1wo ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ? [V2wo_27: $i] :
( ( mem @ V2wo_27 @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Eorderiso @ A_27b @ A_27a ) @ V1wo ) @ V2wo_27 ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2Eallsets__wellorderable,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ? [V1wo: $i] :
( ( mem @ V1wo @ ( ty_2Ewellorder_2Ewellorder @ A_27a ) )
& ( ( ap @ ( c_2Ewellorder_2EelsOf @ A_27a ) @ V1wo )
= V0s ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1P @ V2x ) ) )
=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1P @ V3x ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V4y ) @ V3x ) )
=> ~ ( p @ ( ap @ V1P @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__EQ,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ? [V2x: $i] :
( ( mem @ V2x @ A_27a )
& ( p @ ( ap @ V1P @ V2x ) ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ V1P @ V3x ) )
& ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V4y ) @ V3x ) )
=> ~ ( p @ ( ap @ V1P @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__IND,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
<=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V3y ) @ V2x ) )
=> ( p @ ( ap @ V1P @ V3y ) ) ) )
=> ( p @ ( ap @ V1P @ V2x ) ) ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( p @ ( ap @ V1P @ V4x ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__DCHAIN,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
<=> ~ ? [V1s: $i] :
( ( mem @ V1s @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0_3C_3C @ ( ap @ V1s @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1s @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__UREC,axiom,
! [A_27a: del,A_27b: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
=> ! [V1H: $i] :
( ( mem @ V1H @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V5z ) @ V4x ) )
=> ( ( ap @ V2f @ V5z )
= ( ap @ V3g @ V5z ) ) ) )
=> ( ( ap @ ( ap @ V1H @ V2f ) @ V4x )
= ( ap @ ( ap @ V1H @ V3g ) @ V4x ) ) ) ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( ap @ V6f @ V8x )
= ( ap @ ( ap @ V1H @ V6f ) @ V8x ) ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( ( ap @ V7g @ V9x )
= ( ap @ ( ap @ V1H @ V7g ) @ V9x ) ) ) )
=> ( V6f = V7g ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__UREC__WF,axiom,
! [A_27a: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1H: $i] :
( ( mem @ V1H @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ bool ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V5z ) @ V4x ) )
=> ( ( p @ ( ap @ V2f @ V5z ) )
<=> ( p @ ( ap @ V3g @ V5z ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ V1H @ V2f ) @ V4x ) )
<=> ( p @ ( ap @ ( ap @ V1H @ V3g ) @ V4x ) ) ) ) ) ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ bool ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ V6f @ V8x ) )
<=> ( p @ ( ap @ ( ap @ V1H @ V6f ) @ V8x ) ) ) )
& ! [V9x: $i] :
( ( mem @ V9x @ A_27a )
=> ( ( p @ ( ap @ V7g @ V9x ) )
<=> ( p @ ( ap @ ( ap @ V1H @ V7g ) @ V9x ) ) ) ) )
=> ( V6f = V7g ) ) ) ) ) )
=> ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__REC__INVARIANT,axiom,
! [A_27a: del,A_27b: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
=> ! [V1H: $i] :
( ( mem @ V1H @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V2S: $i] :
( ( mem @ V2S @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
=> ( ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27b ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ! [V6z: $i] :
( ( mem @ V6z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V6z ) @ V5x ) )
=> ( ( ( ap @ V3f @ V6z )
= ( ap @ V4g @ V6z ) )
& ( p @ ( ap @ ( ap @ V2S @ V6z ) @ ( ap @ V3f @ V6z ) ) ) ) ) )
=> ( ( ( ap @ ( ap @ V1H @ V3f ) @ V5x )
= ( ap @ ( ap @ V1H @ V4g ) @ V5x ) )
& ( p @ ( ap @ ( ap @ V2S @ V5x ) @ ( ap @ ( ap @ V1H @ V3f ) @ V5x ) ) ) ) ) ) ) )
=> ? [V7f: $i] :
( ( mem @ V7f @ ( arr @ A_27a @ A_27b ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( ap @ V7f @ V8x )
= ( ap @ ( ap @ V1H @ V7f ) @ V8x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__REC,axiom,
! [A_27a: del,A_27b: del,V0_3C_3C: $i] :
( ( mem @ V0_3C_3C @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0_3C_3C ) )
=> ! [V1H: $i] :
( ( mem @ V1H @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27a @ A_27b ) ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ( ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0_3C_3C @ V5z ) @ V4x ) )
=> ( ( ap @ V2f @ V5z )
= ( ap @ V3g @ V5z ) ) ) )
=> ( ( ap @ ( ap @ V1H @ V2f ) @ V4x )
= ( ap @ ( ap @ V1H @ V3g ) @ V4x ) ) ) ) ) )
=> ? [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( ap @ V6f @ V7x )
= ( ap @ ( ap @ V1H @ V6f ) @ V7x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EWF__num,axiom,
p @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ).
thf(conj_thm_2Ewellorder_2EWF__REC__num,axiom,
! [A_27a: del,V0H: $i] :
( ( mem @ V0H @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ! [V4m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
=> ( ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V4m ) )
= ( ap @ V2g @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) )
=> ( ( ap @ ( ap @ V0H @ V1f ) @ ( inj__ty_2Enum_2Enum @ V3n ) )
= ( ap @ ( ap @ V0H @ V2g ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) )
=> ? [V5f: $i] :
( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
& ! [V6n: tp__ty_2Enum_2Enum] :
( ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) )
= ( ap @ ( ap @ V0H @ V5f ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Efl,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Efl @ A_27a ) @ V0l ) @ V1x ) )
<=> ? [V2y: $i] :
( ( mem @ V2y @ A_27a )
& ( ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V2y ) ) )
| ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2y ) @ V1x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Eposet,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ewellorder_2Eposet @ A_27a ) @ V0l ) )
<=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Efl @ A_27a ) @ V0l ) @ V1x ) )
=> ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V1x ) @ V1x ) ) ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ! [V4z: $i] :
( ( mem @ V4z @ A_27a )
=> ( ( ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) )
& ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V4z ) ) ) )
=> ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V4z ) ) ) ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5x ) @ V6y ) ) )
& ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6y ) @ V5x ) ) ) )
=> ( V5x = V6y ) ) ) ) ) ) ) ).
thf(ax_thm_2Ewellorder_2Echain,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2EChain @ A_27a ) @ V0l ) @ V1P ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( ( p @ ( ap @ V1P @ V2x ) )
& ( p @ ( ap @ V1P @ V3y ) ) )
=> ( ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V2x ) @ V3y ) ) )
| ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V3y ) @ V2x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EZL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( arr @ ( ty_2Epair_2Eprod @ A_27a @ A_27a ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Ewellorder_2Eposet @ A_27a ) @ V0l ) )
& ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p
@ ( ap
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Ewellorder_2Efl @ A_27a ) @ V0l ) @ V2x ) )
@ V1x ) ) )
& ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2EChain @ A_27a ) @ V0l ) @ V3P ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Efl @ A_27a ) @ V0l ) @ V4y ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ V3P @ V5x ) )
=> ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V5x ) @ V4y ) ) ) ) ) ) ) ) )
=> ? [V6y: $i] :
( ( mem @ V6y @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ewellorder_2Efl @ A_27a ) @ V0l ) @ V6y ) )
& ! [V7x: $i] :
( ( mem @ V7x @ A_27a )
=> ( ( p @ ( ap @ V0l @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27a ) @ V6y ) @ V7x ) ) )
=> ( V6y = V7x ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EPOSET__RESTRICTED__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( p
@ ( ap @ ( c_2Ewellorder_2Eposet @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) @ bool )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1x: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V2y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EFL__RESTRICTED__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ap @ ( c_2Ewellorder_2Efl @ ( arr @ A_27a @ bool ) )
@ ( ap @ ( c_2Epair_2EUNCURRY @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) @ bool )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1x: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V2y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V1x ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V2y ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1x ) @ V2y ) ) ) ) ) ) )
= V0P ) ) ).
thf(conj_thm_2Ewellorder_2EZL__SUBSETS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V1c ) )
=> ( p @ ( ap @ V0P @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ bool ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3x ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4y ) @ V1c ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3x ) @ V4y ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V4y ) @ V3x ) ) ) ) ) ) )
=> ? [V5z: $i] :
( ( mem @ V5z @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V5z ) )
& ! [V6x: $i] :
( ( mem @ V6x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V6x ) @ V1c ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V6x ) @ V5z ) ) ) ) ) ) )
=> ? [V7a: $i] :
( ( mem @ V7a @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V7a ) )
& ! [V8x: $i] :
( ( mem @ V8x @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ V8x ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V7a ) @ V8x ) ) )
=> ( V7a = V8x ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EZL__SUBSETS__BIGUNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V1c: $i] :
( ( mem @ V1c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ! [V2x: $i] :
( ( mem @ V2x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2x ) @ V1c ) )
=> ( p @ ( ap @ V0P @ V2x ) ) ) )
& ! [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ bool ) )
=> ! [V4y: $i] :
( ( mem @ V4y @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3x ) @ V1c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4y ) @ V1c ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3x ) @ V4y ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V4y ) @ V3x ) ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1c ) ) ) ) )
=> ? [V5a: $i] :
( ( mem @ V5a @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V5a ) )
& ! [V6x: $i] :
( ( mem @ V6x @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ V6x ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V5a ) @ V6x ) ) )
=> ( V5a = V6x ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EZL__SUBSETS__BIGUNION__NONEMPTY,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ? [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V1x ) ) )
& ! [V2c: $i] :
( ( mem @ V2c @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ? [V3x: $i] :
( ( mem @ V3x @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3x ) @ V2c ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4x ) @ V2c ) )
=> ( p @ ( ap @ V0P @ V4x ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ ( arr @ A_27a @ bool ) )
=> ! [V6y: $i] :
( ( mem @ V6y @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V5x ) @ V2c ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V6y ) @ V2c ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V5x ) @ V6y ) )
| ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V6y ) @ V5x ) ) ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V2c ) ) ) ) ) )
=> ? [V7a: $i] :
( ( mem @ V7a @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V0P @ V7a ) )
& ! [V8x: $i] :
( ( mem @ V8x @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ V8x ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V7a ) @ V8x ) ) )
=> ( V7a = V8x ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EFLATTEN__LEMMA,axiom,
! [A_27a: del,A_27b: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ A_27b @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( ( ap @ V0g @ ( ap @ V1f @ V3x ) )
= V3x ) ) )
<=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V2s ) )
& ( V4y
= ( ap @ V1f @ V5x ) ) )
=> ( ( ap @ V0g @ V4y )
= V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Ewellorder_2ETARSKI__SET,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ V0f @ V1s ) ) @ ( ap @ V0f @ V2t ) ) ) ) ) )
=> ? [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
& ( ( ap @ V0f @ V3s )
= V3s ) ) ) ) ).
thf(conj_thm_2Ewellorder_2EINJECTIVE__LEFT__INVERSE__NONEMPTY,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27b @ bool ) )
=> ( ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5y ) @ V1s ) )
& ( ( ap @ V0f @ V4x )
= ( ap @ V0f @ V5y ) ) )
=> ( V4x = V5y ) ) ) )
<=> ? [V6g: $i] :
( ( mem @ V6g @ ( arr @ A_27b @ A_27a ) )
& ! [V7y: $i] :
( ( mem @ V7y @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V7y ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V6g @ V7y ) ) @ V1s ) ) ) )
& ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V8x ) @ V1s ) )
=> ( ( ap @ V6g @ ( ap @ V0f @ V8x ) )
= V8x ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------