ITP001 Axioms: ITP088^5.ax
%------------------------------------------------------------------------------
% File : ITP088^5 : TPTP v9.0.0. 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 : quotient_pred_set^2.ax [Gau20]
% : HL4088^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 64 ( 0 unt; 8 typ; 0 def)
% Number of atoms : 2146 ( 12 equ; 0 cnn)
% Maximal formula atoms : 118 ( 33 avg)
% Number of connectives : 3635 ( 3 ~; 1 |; 13 &;3320 @)
% ( 24 <=>; 274 =>; 0 <=; 0 <~>)
% Maximal formula depth : 42 ( 22 avg;3320 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 10 ( 10 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 46 usr; 38 con; 0-2 aty)
% Number of variables : 346 ( 20 ^ 326 !; 0 ?; 346 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Equotient__pred__set_2EDELETER,type,
c_2Equotient__pred__set_2EDELETER: del > $i ).
thf(mem_c_2Equotient__pred__set_2EDELETER,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EDISJOINTR,type,
c_2Equotient__pred__set_2EDISJOINTR: del > $i ).
thf(mem_c_2Equotient__pred__set_2EDISJOINTR,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EFINITER,type,
c_2Equotient__pred__set_2EFINITER: del > $i ).
thf(mem_c_2Equotient__pred__set_2EFINITER,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EGSPECR,type,
c_2Equotient__pred__set_2EGSPECR: del > del > $i ).
thf(mem_c_2Equotient__pred__set_2EGSPECR,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) @ ( arr @ A_27b @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EIMAGER,type,
c_2Equotient__pred__set_2EIMAGER: del > del > $i ).
thf(mem_c_2Equotient__pred__set_2EIMAGER,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27b @ bool ) ) ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EINSERTR,type,
c_2Equotient__pred__set_2EINSERTR: del > $i ).
thf(mem_c_2Equotient__pred__set_2EINSERTR,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2EPSUBSETR,type,
c_2Equotient__pred__set_2EPSUBSETR: del > $i ).
thf(mem_c_2Equotient__pred__set_2EPSUBSETR,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Equotient__pred__set_2ESUBSETR,type,
c_2Equotient__pred__set_2ESUBSETR: del > $i ).
thf(mem_c_2Equotient__pred__set_2ESUBSETR,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__SET__MAP,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27b @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V0f ) @ ( c_2Ecombin_2EI @ bool ) ) @ V1s ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V2x ) ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2ESET__REL,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V1s ) @ V2t ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4y ) @ V2t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2ESET__REL__MP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27a @ bool ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s ) @ V4t ) )
& ( p @ ( ap @ ( ap @ V0R @ V5x ) @ V6y ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V6y ) @ V4t ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27b )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V3x ) @ V4s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ ( ap @ V2rep @ V3x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x1: $i] :
( ( mem @ V3x1 @ A_27a )
=> ! [V4x2: $i] :
( ( mem @ V4x2 @ A_27a )
=> ! [V5s1: $i] :
( ( mem @ V5s1 @ ( arr @ A_27a @ bool ) )
=> ! [V6s2: $i] :
( ( mem @ V6s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x1 ) @ V4x2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5s1 ) @ V6s2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x1 ) @ V5s1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x2 ) @ V6s2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EGSPECR__def,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2f ) @ V3v ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ bool @ A_27b @ bool ) @ V1R2 ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V3v ) @ c_2Ebool_2ET ) ) @ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__GSPECR,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27b @ ( ty_2Epair_2Eprod @ A_27a @ bool ) ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3v ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EGSPECR @ A_27b @ A_27a ) @ V0R1 ) @ V1R2 ) @ V2f ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27b @ bool ) @ V0R1 ) )
@ ( lam @ A_27b
@ ^ [V4x: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27a @ bool @ A_27a @ bool ) @ V1R2 ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3v ) @ c_2Ebool_2ET ) ) @ ( ap @ V2f @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EGSPEC__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27c @ ( ty_2Epair_2Eprod @ A_27d @ bool ) ) )
=> ( ( ap @ ( c_2Epred__set_2EGSPEC @ A_27d @ A_27c ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27d @ bool @ A_27b @ bool ) @ V5rep2 ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( ty_2Epair_2Eprod @ A_27d @ bool ) @ A_27c @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) @ V1abs1 ) @ ( ap @ ( ap @ ( c_2Epair_2E_23_23 @ A_27d @ bool @ A_27b @ bool ) @ V5rep2 ) @ ( c_2Ecombin_2EI @ bool ) ) ) @ V6f ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EGSPECR__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) )
=> ! [V7f2: $i] :
( ( mem @ V7f2 @ ( arr @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( ty_2Epair_2Eprod @ A_27b @ bool ) ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Equotient__pair_2E_23_23_23 @ A_27b @ bool @ A_27b @ bool ) @ V3R2 ) @ ( c_2Emin_2E_3D @ bool ) ) ) @ V6f1 ) @ V7f2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ bool ) @ V3R2 ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EGSPECR @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V7f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EEMPTY__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( ( c_2Epred__set_2EEMPTY @ A_27b )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EEMPTY__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EUNIV__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( ( c_2Epred__set_2EUNIV @ A_27b )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EUNIV__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EUNION__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27b ) @ V3s ) @ V4t )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EUNION__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V3s1 ) @ V5t1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EINTER__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27b ) @ V3s ) @ V4t )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EINTER__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V3s1 ) @ V5t1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2ESUBSETR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ V0R ) @ V1s ) @ V2t ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2ESUBSET__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27b ) @ V3s ) @ V4t ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2ESUBSETR__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ V0R ) @ V3s1 ) @ V5t1 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ V0R ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EPSUBSETR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a ) @ V0R ) @ V1s ) @ V2t ) )
<=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2ESUBSETR @ A_27a ) @ V0R ) @ V1s ) @ V2t ) )
& ~ ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EPSUBSET__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EPSUBSET @ A_27b ) @ V3s ) @ V4t ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EPSUBSETR__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a ) @ V0R ) @ V3s1 ) @ V5t1 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EPSUBSETR @ A_27a ) @ V0R ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EINSERTR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V1x ) @ V2s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3y ) @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ V0R @ V3y ) @ V1x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V2s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__INSERTR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V1x ) @ V2s ) ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V1x ) )
| ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EINSERT__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27b ) @ V4x ) @ V3s )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ ( ap @ V2rep @ V4x ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EINSERTR__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3x1: $i] :
( ( mem @ V3x1 @ A_27a )
=> ! [V4x2: $i] :
( ( mem @ V4x2 @ A_27a )
=> ! [V5s1: $i] :
( ( mem @ V5s1 @ ( arr @ A_27a @ bool ) )
=> ! [V6s2: $i] :
( ( mem @ V6s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x1 ) @ V4x2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5s1 ) @ V6s2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V3x1 ) @ V5s1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V4x2 ) @ V6s2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EDELETER__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ V0R ) @ V1s ) @ V2x )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V3y ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__DELETER,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ V0R ) @ V1s ) @ V2x ) ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ V1s ) )
& ~ ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDELETE__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4x: $i] :
( ( mem @ V4x @ A_27b )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDELETE @ A_27b ) @ V3s ) @ V4x )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ V2rep @ V4x ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDELETER__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5x1: $i] :
( ( mem @ V5x1 @ A_27a )
=> ! [V6x2: $i] :
( ( mem @ V6x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V5x1 ) @ V6x2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ V0R ) @ V3s1 ) @ V5x1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDELETER @ A_27a ) @ V0R ) @ V4s2 ) @ V6x2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDIFF__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27b ) @ V3s ) @ V4t )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ bool @ A_27a @ bool ) @ V2rep ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDIFF__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V3s1 ) @ V5t1 ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EDISJOINTR__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a ) @ V0R ) @ V1s ) @ V2t ) )
<=> ~ ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDISJOINT__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ! [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2EDISJOINT @ A_27b ) @ V3s ) @ V4t ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4t ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EDISJOINTR__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ! [V5t1: $i] :
( ( mem @ V5t1 @ ( arr @ A_27a @ bool ) )
=> ! [V6t2: $i] :
( ( mem @ V6t2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V5t1 ) @ V6t2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a ) @ V0R ) @ V3s1 ) @ V5t1 ) )
<=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EDISJOINTR @ A_27a ) @ V0R ) @ V4s2 ) @ V6t2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EFINITER__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V1s ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ bool )
@ ^ [V2P: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V2P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V3s: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ V2P @ V3s ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V4e: $i] : ( ap @ V2P @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V4e ) @ V3s ) ) ) ) ) ) ) ) )
@ ( ap @ V2P @ V1s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITE__PRS,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ V3s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITER__EQ,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1s1: $i] :
( ( mem @ V1s1 @ ( arr @ A_27a @ bool ) )
=> ! [V2s2: $i] :
( ( mem @ V2s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V1s1 ) @ V2s2 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V1s1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V2s2 ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITER__RSP,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs: $i] :
( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
=> ! [V2rep: $i] :
( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
=> ! [V3s1: $i] :
( ( mem @ V3s1 @ ( arr @ A_27a @ bool ) )
=> ! [V4s2: $i] :
( ( mem @ V4s2 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ V3s1 ) @ V4s2 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V3s1 ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V4s2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITER__EMPTY,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITER__INSERTR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1s: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V1s ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V2x ) @ V1s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EABSORPTIONR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ bool ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ V2s ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V1x ) @ V2s ) ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EFINITER__INDUCT,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ ( arr @ A_27a @ bool ) @ bool )
@ ^ [V1P: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2s: $i] :
( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V2s ) ) @ ( ap @ V1P @ V2s ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3e: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3e ) @ V2s ) ) ) @ ( ap @ V1P @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EINSERTR @ A_27a ) @ V0R ) @ V3e ) @ V2s ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ bool ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R ) @ ( c_2Emin_2E_3D @ bool ) ) ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V4s: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EFINITER @ A_27a ) @ V0R ) @ V4s ) ) @ ( ap @ V1P @ V4s ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient__pred__set_2EIMAGER__def,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27b ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2f ) @ V3s )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27b @ A_27b )
@ ( lam @ A_27b
@ ^ [V4y: $i] :
( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27b @ bool ) @ V4y )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V1R2 @ V4y ) @ ( ap @ V2f @ V5x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V3s ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIN__IMAGER,axiom,
! [A_27a: del,A_27b: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27b )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ A_27b ) )
=> ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2y ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V3f ) @ V4s ) ) )
<=> ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) )
@ ( lam @ A_27a
@ ^ [V5x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ V1R2 @ V2y ) @ ( ap @ V3f @ V5x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ V4s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIMAGE__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27c @ A_27d ) )
=> ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27c @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27c @ A_27d ) @ V6f ) @ V7s )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27d @ bool @ A_27b @ bool ) @ V5rep2 ) @ ( c_2Ecombin_2EI @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27c @ bool ) @ V1abs1 ) @ ( c_2Ecombin_2EI @ bool ) ) @ V7s ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient__pred__set_2EIMAGER__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V7f2: $i] :
( ( mem @ V7f2 @ ( arr @ A_27a @ A_27b ) )
=> ! [V8s1: $i] :
( ( mem @ V8s1 @ ( arr @ A_27a @ bool ) )
=> ! [V9s2: $i] :
( ( mem @ V9s2 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f1 ) @ V7f2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ bool ) @ V0R1 ) @ ( c_2Emin_2E_3D @ bool ) ) @ V8s1 ) @ V9s2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ bool ) @ V3R2 ) @ ( c_2Emin_2E_3D @ bool ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f1 ) @ V8s1 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient__pred__set_2EIMAGER @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V7f2 ) @ V9s2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------