ITP001 Axioms: ITP050^5.ax
%------------------------------------------------------------------------------
% File : ITP050^5 : 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^2.ax [Gau20]
% : HL4050^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 115 ( 4 unt; 8 typ; 0 def)
% Number of atoms : 2967 ( 38 equ; 0 cnn)
% Maximal formula atoms : 71 ( 25 avg)
% Number of connectives : 6212 ( 2 ~; 2 |; 57 &;5401 @)
% ( 35 <=>; 715 =>; 0 <=; 0 <~>)
% Maximal formula depth : 54 ( 22 avg;5401 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 14 ( 14 >; 0 *; 0 +; 0 <<)
% Number of symbols : 35 ( 34 usr; 26 con; 0-4 aty)
% Number of variables : 784 ( 16 ^ 767 !; 1 ?; 784 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Equotient_2E_2D_2D_3E,type,
c_2Equotient_2E_2D_2D_3E: del > del > del > del > $i ).
thf(mem_c_2Equotient_2E_2D_2D_3E,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ A_27a @ A_27d ) ) ) ) ) ).
thf(tp_c_2Equotient_2E_3D_3D_3D_3E,type,
c_2Equotient_2E_3D_3D_3D_3E: del > del > $i ).
thf(mem_c_2Equotient_2E_3D_3D_3D_3E,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2E_3D_3D_3D_3E @ 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 @ A_27b ) @ bool ) ) ) ) ) ).
thf(tp_c_2Equotient_2E_3F_21_21,type,
c_2Equotient_2E_3F_21_21: del > $i ).
thf(mem_c_2Equotient_2E_3F_21_21,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2E_3F_21_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).
thf(tp_c_2Equotient_2EEQUIV,type,
c_2Equotient_2EEQUIV: del > $i ).
thf(mem_c_2Equotient_2EEQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2EEQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Equotient_2EPARTIAL__EQUIV,type,
c_2Equotient_2EPARTIAL__EQUIV: del > $i ).
thf(mem_c_2Equotient_2EPARTIAL__EQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).
thf(tp_c_2Equotient_2EQUOTIENT,type,
c_2Equotient_2EQUOTIENT: del > del > $i ).
thf(mem_c_2Equotient_2EQUOTIENT,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ) ).
thf(tp_c_2Equotient_2ERES__EXISTS__EQUIV,type,
c_2Equotient_2ERES__EXISTS__EQUIV: del > $i ).
thf(mem_c_2Equotient_2ERES__EXISTS__EQUIV,axiom,
! [A_27a: del] : ( mem @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Equotient_2Erespects,type,
c_2Equotient_2Erespects: del > del > $i ).
thf(mem_c_2Equotient_2Erespects,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2Erespects @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(ax_thm_2Equotient_2EEQUIV__def,axiom,
! [A_27a: del,V0E: $i] :
( ( mem @ V0E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0E ) )
<=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0E @ V1x ) @ V2y ) )
<=> ( ( ap @ V0E @ V1x )
= ( ap @ V0E @ V2y ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EPARTIAL__EQUIV__def,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ V0R ) )
<=> ( ? [V1x: $i] :
( ( mem @ V1x @ A_27a )
& ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) )
& ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ! [V3y: $i] :
( ( mem @ V3y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V3y ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V2x ) @ V2x ) )
& ( p @ ( ap @ ( ap @ V0R @ V3y ) @ V3y ) )
& ( ( ap @ V0R @ V2x )
= ( ap @ V0R @ V3y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__IMP__PARTIAL__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0R ) )
=> ( p @ ( ap @ ( c_2Equotient_2EPARTIAL__EQUIV @ A_27a ) @ V0R ) ) ) ) ).
thf(ax_thm_2Equotient_2EQUOTIENT__def,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 ) )
<=> ( ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ap @ V1abs @ ( ap @ V2rep @ V3a ) )
= V3a ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27b )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V4a ) ) @ ( ap @ V2rep @ V4a ) ) ) )
& ! [V5r: $i] :
( ( mem @ V5r @ A_27a )
=> ! [V6s: $i] :
( ( mem @ V6s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V5r ) @ V6s ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V5r ) @ V5r ) )
& ( p @ ( ap @ ( ap @ V0R @ V6s ) @ V6s ) )
& ( ( ap @ V1abs @ V5r )
= ( ap @ V1abs @ V6s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__ABS__REP,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 ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( ( ap @ V1abs @ ( ap @ V2rep @ V3a ) )
= V3a ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REP__REFL,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 ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V3a ) ) @ ( ap @ V2rep @ V3a ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL,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 ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
<=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
& ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
& ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__ABS,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 ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
=> ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__ABS__EQ,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 ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ! [V4s: $i] :
( ( mem @ V4s @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
<=> ( ( ap @ V1abs @ V3r )
= ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REL__REP,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 ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27b )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V3a ) ) @ ( ap @ V2rep @ V4b ) ) )
<=> ( V3a = V4b ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__REP__ABS,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 ) )
=> ! [V3r: $i] :
( ( mem @ V3r @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ ( ap @ V1abs @ V3r ) ) ) @ V3r ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EIDENTITY__EQUIV,axiom,
! [A_27a: del] : ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) ) ).
thf(conj_thm_2Equotient_2EIDENTITY__QUOTIENT,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27a ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27a ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__REFL__SYM__TRANS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V2y ) )
<=> ( ( ap @ V0R @ V1x )
= ( ap @ V0R @ V2y ) ) ) ) )
<=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V3x ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V5y ) @ V4x ) ) ) ) )
& ! [V6x: $i] :
( ( mem @ V6x @ A_27a )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ! [V8z: $i] :
( ( mem @ V8z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V6x ) @ V7y ) )
& ( p @ ( ap @ ( ap @ V0R @ V7y ) @ V8z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V6x ) @ V8z ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__SYM,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_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V3x ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EQUOTIENT__TRANS,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_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ! [V5z: $i] :
( ( mem @ V5z @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) )
& ( p @ ( ap @ ( ap @ V0R @ V4y ) @ V5z ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ V3x ) @ V5z ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EFUN__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ( ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g )
= ( lam @ ( arr @ A_27c @ A_27b )
@ ^ [V2h: $i] :
( lam @ A_27a
@ ^ [V3x: $i] : ( ap @ V1g @ ( ap @ V2h @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__MAP__THM,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27c ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ A_27b @ A_27d ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ A_27c @ A_27b ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ V0f ) @ V1g ) @ V2h ) @ V3x )
= ( ap @ V1g @ ( ap @ V2h @ ( ap @ V0f @ V3x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__MAP__I,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27a @ A_27b ) @ ( c_2Ecombin_2EI @ A_27a ) ) @ ( c_2Ecombin_2EI @ A_27b ) )
= ( c_2Ecombin_2EI @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient_2EIN__FUN,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ bool @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27b @ bool ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V0f ) @ V1g ) @ V2s ) ) )
<=> ( p @ ( ap @ V1g @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ ( ap @ V0f @ V3x ) ) @ V2s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2EFUN__REL,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 ) )
=> ! [V3g: $i] :
( ( mem @ V3g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) @ V2f ) @ V3g ) )
<=> ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V4x ) @ V5y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V4x ) ) @ ( ap @ V3g @ V5y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQ,axiom,
! [A_27a: del,A_27b: del] :
( ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ ( c_2Emin_2E_3D @ A_27a ) ) @ ( c_2Emin_2E_3D @ A_27b ) )
= ( c_2Emin_2E_3D @ ( arr @ A_27a @ A_27b ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__QUOTIENT,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 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2Erespects__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Equotient_2Erespects @ A_27a @ A_27b )
= ( c_2Ecombin_2EW @ A_27a @ A_27b ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) @ V1x ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EIN__RESPECTS,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V1x ) @ V1x ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__THM,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 ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2f ) )
<=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V3x ) @ V4y ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__MP,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 ) )
=> ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V2f ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V3x ) @ V4y ) ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f @ V3x ) ) @ ( ap @ V2f @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__REP__ABS,axiom,
! [A_27a: del,A_27b: del,A_27c: 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 ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V4f ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V5x ) @ V5x ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V4f @ ( ap @ V2rep1 @ ( ap @ V1abs1 @ V5x ) ) ) ) @ ( ap @ V4f @ V5x ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERESPECTS__o,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1R2: $i] :
( ( mem @ V1R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V2R3: $i] :
( ( mem @ V2R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ A_27c ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27b @ A_27c ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V1R2 ) @ V2R3 ) ) @ V3f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V1R2 ) ) @ V4g ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27c ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27c ) @ V0R1 ) @ V2R3 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V3f ) @ V4g ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Equotient_2E_3F_21_21,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2E_3F_21_21 @ A_27a ) @ V0P ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V0P ) ) ) ) ).
thf(ax_thm_2Equotient_2ERES__EXISTS__EQUIV__DEF,axiom,
! [A_27a: del] :
( ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a )
= ( lam @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) )
@ ^ [V0R: $i] :
( lam @ ( arr @ A_27a @ bool )
@ ^ [V1P: $i] :
( ap
@ ( ap @ c_2Ebool_2E_2F_5C
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1P @ V2x ) ) ) )
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V3x: $i] :
( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V3x ) ) @ ( ap @ V1P @ V4y ) ) ) @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__EQUIV,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V1m ) )
<=> ( ( p
@ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ V1m @ V2x ) ) ) )
& ( 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_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) )
@ ( lam @ A_27a
@ ^ [V4y: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1m @ V3x ) ) @ ( ap @ V1m @ V4y ) ) ) @ ( ap @ ( ap @ V0R @ V3x ) @ V4y ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQ__REL,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_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__MP,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_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V6f @ V8x ) ) @ ( ap @ V7g @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__EQUALS,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_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) ) )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) )
<=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V6f @ V8x ) ) @ ( ap @ V7g @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFUN__REL__IMP,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_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) )
& ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V6f @ V8x ) ) @ ( ap @ V7g @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUALS__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 )
=> ! [V4y: $i] :
( ( mem @ V4y @ A_27b )
=> ( ( V3x = V4y )
<=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ V2rep @ V3x ) ) @ ( ap @ V2rep @ V4y ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUALS__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 )
=> ! [V5y1: $i] :
( ( mem @ V5y1 @ A_27a )
=> ! [V6y2: $i] :
( ( mem @ V6y2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V0R @ V3x1 ) @ V4x2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V5y1 ) @ V6y2 ) ) )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3x1 ) @ V5y1 ) )
<=> ( p @ ( ap @ ( ap @ V0R @ V4x2 ) @ V6y2 ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELAMBDA__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 ) )
=> ( ( lam @ A_27c
@ ^ [V7x: $i] : ( ap @ V6f @ V7x ) )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ V5rep2 @ ( ap @ V6f @ ( ap @ V1abs1 @ V8x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELAMBDA__PRS1,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 ) )
=> ( ( lam @ A_27c
@ ^ [V7x: $i] : ( ap @ V6f @ V7x ) )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) @ V8x ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELAMBDA__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 ) )
=> ( ( 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 @ A_27b ) @ V0R1 ) @ V3R2 )
@ ( lam @ A_27a
@ ^ [V8x: $i] : ( ap @ V6f1 @ V8x ) ) )
@ ( lam @ A_27a
@ ^ [V9y: $i] : ( ap @ V7f2 @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EABSTRACT__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 ) )
=> ( V6f
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__ABSTRACT__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 ) )
=> ( ( 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 @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V6f1 ) ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V7f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELET__RES__ABSTRACT,axiom,
! [A_27a: del,A_27b: del,V0r: $i] :
( ( mem @ V0r @ ( arr @ A_27a @ bool ) )
=> ! [V1lam: $i] :
( ( mem @ V1lam @ ( arr @ A_27a @ A_27b ) )
=> ! [V2v: $i] :
( ( mem @ V2v @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2v ) @ V0r ) )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ V0r ) @ V1lam ) ) @ V2v )
= ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27b ) @ V1lam ) @ V2v ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELAMBDA__REP__ABS__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0REL1: $i] :
( ( mem @ V0REL1 @ ( 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 ) )
=> ! [V3REL2: $i] :
( ( mem @ V3REL2 @ ( 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 ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V7f2: $i] :
( ( mem @ V7f2 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ! [V8r: $i] :
( ( mem @ V8r @ A_27a )
=> ! [V9r_27: $i] :
( ( mem @ V9r_27 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0REL1 @ V8r ) @ V9r_27 ) )
=> ( p @ ( ap @ ( ap @ V0REL1 @ V8r ) @ ( ap @ V2rep1 @ ( ap @ V1abs1 @ V9r_27 ) ) ) ) ) ) )
& ! [V10r: $i] :
( ( mem @ V10r @ A_27b )
=> ! [V11r_27: $i] :
( ( mem @ V11r_27 @ A_27b )
=> ( ( p @ ( ap @ ( ap @ V3REL2 @ V10r ) @ V11r_27 ) )
=> ( p @ ( ap @ ( ap @ V3REL2 @ V10r ) @ ( ap @ V5rep2 @ ( ap @ V4abs2 @ V11r_27 ) ) ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0REL1 ) @ V3REL2 ) @ V6f1 ) @ V7f2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0REL1 ) @ V3REL2 ) @ V6f1 ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7f2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EREP__ABS__RSP,axiom,
! [A_27a: del,A_27b: del,V0REL: $i] :
( ( mem @ V0REL @ ( 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 ) @ V0REL ) @ V1abs ) @ V2rep ) )
=> ! [V3x1: $i] :
( ( mem @ V3x1 @ A_27a )
=> ! [V4x2: $i] :
( ( mem @ V4x2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0REL @ V3x1 ) @ V4x2 ) )
=> ( p @ ( ap @ ( ap @ V0REL @ V3x1 ) @ ( ap @ V2rep @ ( ap @ V1abs @ V4x2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFORALL__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27b ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__FORALL__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ bool ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( 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 ) ) @ V3f ) @ V4g ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V4g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__FORALL__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 ) )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27b ) @ V3P ) @ V4f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3P ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEXISTS__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27b ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ bool ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( 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 ) ) @ V3f ) @ V4g ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V4g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__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 ) )
=> ! [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27b @ bool ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27b ) @ V3P ) @ V4f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3P ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V4f ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEXISTS__UNIQUE__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27b @ bool ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27b ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ bool @ A_27b @ bool ) @ V1abs ) @ ( c_2Ecombin_2EI @ bool ) ) @ V3f ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__EQUIV__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 ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ A_27a @ bool ) )
=> ! [V4g: $i] :
( ( mem @ V4g @ ( 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 ) ) @ V3f ) @ V4g ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V3f ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V4g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ECOND__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 ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ bool )
=> ! [V4b: $i] :
( ( mem @ V4b @ A_27b )
=> ! [V5c: $i] :
( ( mem @ V5c @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27b ) @ V3a ) @ V4b ) @ V5c )
= ( ap @ V1abs @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V3a ) @ ( ap @ V2rep @ V4b ) ) @ ( ap @ V2rep @ V5c ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ECOND__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 ) )
=> ! [V3a1: $i] :
( ( mem @ V3a1 @ bool )
=> ! [V4a2: $i] :
( ( mem @ V4a2 @ bool )
=> ! [V5b1: $i] :
( ( mem @ V5b1 @ A_27a )
=> ! [V6b2: $i] :
( ( mem @ V6b2 @ A_27a )
=> ! [V7c1: $i] :
( ( mem @ V7c1 @ A_27a )
=> ! [V8c2: $i] :
( ( mem @ V8c2 @ A_27a )
=> ( ( ( ( p @ V3a1 )
<=> ( p @ V4a2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V5b1 ) @ V6b2 ) )
& ( p @ ( ap @ ( ap @ V0R @ V7c1 ) @ V8c2 ) ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V3a1 ) @ V5b1 ) @ V7c1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ V4a2 ) @ V6b2 ) @ V8c2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELET__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 ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27c )
=> ( ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27c @ A_27d ) @ V6f ) @ V7x )
= ( ap @ V4abs2 @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) ) @ ( ap @ V2rep1 @ V7x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELET__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 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27b ) @ V6f ) @ V8x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2ELET @ A_27a @ A_27b ) @ V7g ) @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2Eliteral__case__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 ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27c )
=> ( ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27c @ A_27d ) @ V6f ) @ V7x )
= ( ap @ V4abs2 @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) ) @ ( ap @ V2rep1 @ V7x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2Eliteral__case__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 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b ) @ V6f ) @ V8x ) ) @ ( ap @ ( ap @ ( c_2Ebool_2Eliteral__case @ A_27a @ A_27b ) @ V7g ) @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EAPPLY__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 ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27c )
=> ( ( ap @ V6f @ V7x )
= ( ap @ V4abs2 @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V6f ) @ ( ap @ V2rep1 @ V7x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EAPPLY__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 ) )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
=> ! [V7g: $i] :
( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
=> ! [V8x: $i] :
( ( mem @ V8x @ A_27a )
=> ! [V9y: $i] :
( ( mem @ V9y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V8x ) @ V9y ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ V6f @ V8x ) ) @ ( ap @ V7g @ V9y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EI__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 ) )
=> ! [V3e: $i] :
( ( mem @ V3e @ A_27b )
=> ( ( ap @ ( c_2Ecombin_2EI @ A_27b ) @ V3e )
= ( ap @ V1abs @ ( ap @ ( c_2Ecombin_2EI @ A_27a ) @ ( ap @ V2rep @ V3e ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EI__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 ) )
=> ! [V3e1: $i] :
( ( mem @ V3e1 @ A_27a )
=> ! [V4e2: $i] :
( ( mem @ V4e2 @ A_27a )
=> ( ( p @ ( ap @ ( ap @ V0R @ V3e1 ) @ V4e2 ) )
=> ( p @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Ecombin_2EI @ A_27a ) @ V3e1 ) ) @ ( ap @ ( c_2Ecombin_2EI @ A_27a ) @ V4e2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EK__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 ) )
=> ! [V6x: $i] :
( ( mem @ V6x @ A_27c )
=> ! [V7y: $i] :
( ( mem @ V7y @ A_27d )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2EK @ A_27c @ A_27d ) @ V6x ) @ V7y )
= ( ap @ V1abs1 @ ( ap @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ ( ap @ V2rep1 @ V6x ) ) @ ( ap @ V5rep2 @ V7y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EK__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 ) )
=> ! [V6x1: $i] :
( ( mem @ V6x1 @ A_27a )
=> ! [V7x2: $i] :
( ( mem @ V7x2 @ A_27a )
=> ! [V8y1: $i] :
( ( mem @ V8y1 @ A_27b )
=> ! [V9y2: $i] :
( ( mem @ V9y2 @ A_27b )
=> ( ( ( p @ ( ap @ ( ap @ V0R1 @ V6x1 ) @ V7x2 ) )
& ( p @ ( ap @ ( ap @ V3R2 @ V8y1 ) @ V9y2 ) ) )
=> ( p @ ( ap @ ( ap @ V0R1 @ ( ap @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ V6x1 ) @ V8y1 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EK @ A_27a @ A_27b ) @ V7x2 ) @ V9y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2Eo__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27e @ A_27f ) )
=> ! [V10g: $i] :
( ( mem @ V10g @ ( arr @ A_27d @ A_27e ) )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27d @ A_27f @ A_27e ) @ V9f ) @ V10g )
= ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27d @ A_27c @ A_27a @ A_27f ) @ V2rep1 ) @ V7abs3 ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c ) @ V4abs2 ) @ V8rep3 ) @ V9f ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27e @ A_27d @ A_27b ) @ V1abs1 ) @ V5rep2 ) @ V10g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2Eo__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ A_27b @ A_27c ) )
=> ! [V10f2: $i] :
( ( mem @ V10f2 @ ( arr @ A_27b @ A_27c ) )
=> ! [V11g1: $i] :
( ( mem @ V11g1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V12g2: $i] :
( ( mem @ V12g2 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V3R2 ) @ V6R3 ) @ V9f1 ) @ V10f2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V11g1 ) @ V12g2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27c ) @ V0R1 ) @ V6R3 ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V9f1 ) @ V11g1 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ A_27a @ A_27c @ A_27b ) @ V10f2 ) @ V12g2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EC__PRS,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ A_27d @ ( arr @ A_27e @ A_27f ) ) )
=> ! [V10x: $i] :
( ( mem @ V10x @ A_27e )
=> ! [V11y: $i] :
( ( mem @ V11y @ A_27d )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27d @ A_27e @ A_27f ) @ V9f ) @ V10x ) @ V11y )
= ( ap @ V7abs3 @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( arr @ A_27e @ A_27f ) @ A_27d @ ( arr @ A_27b @ A_27c ) ) @ V1abs1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27b @ A_27f @ A_27e @ A_27c ) @ V4abs2 ) @ V8rep3 ) ) @ V9f ) ) @ ( ap @ V5rep2 @ V10x ) ) @ ( ap @ V2rep1 @ V11y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EC__RSP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,A_27e: del,A_27f: del,V0R1: $i] :
( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1abs1: $i] :
( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27d ) )
=> ! [V2rep1: $i] :
( ( mem @ V2rep1 @ ( arr @ A_27d @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27d ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
=> ! [V3R2: $i] :
( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
=> ! [V4abs2: $i] :
( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27e ) )
=> ! [V5rep2: $i] :
( ( mem @ V5rep2 @ ( arr @ A_27e @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27e ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
=> ! [V6R3: $i] :
( ( mem @ V6R3 @ ( arr @ A_27c @ ( arr @ A_27c @ bool ) ) )
=> ! [V7abs3: $i] :
( ( mem @ V7abs3 @ ( arr @ A_27c @ A_27f ) )
=> ! [V8rep3: $i] :
( ( mem @ V8rep3 @ ( arr @ A_27f @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27c @ A_27f ) @ V6R3 ) @ V7abs3 ) @ V8rep3 ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V10f2: $i] :
( ( mem @ V10f2 @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) )
=> ! [V11x1: $i] :
( ( mem @ V11x1 @ A_27b )
=> ! [V12x2: $i] :
( ( mem @ V12x2 @ A_27b )
=> ! [V13y1: $i] :
( ( mem @ V13y1 @ A_27a )
=> ! [V14y2: $i] :
( ( mem @ V14y2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( arr @ A_27b @ A_27c ) ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27b @ A_27c ) @ V3R2 ) @ V6R3 ) ) @ V9f1 ) @ V10f2 ) )
& ( p @ ( ap @ ( ap @ V3R2 @ V11x1 ) @ V12x2 ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V13y1 ) @ V14y2 ) ) )
=> ( p @ ( ap @ ( ap @ V6R3 @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ V9f1 ) @ V11x1 ) @ V13y1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ A_27a @ A_27b @ A_27c ) @ V10f2 ) @ V12x2 ) @ V14y2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EW__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 @ ( arr @ A_27c @ A_27d ) ) )
=> ! [V7x: $i] :
( ( mem @ V7x @ A_27c )
=> ( ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27c @ A_27d ) @ V6f ) @ V7x )
= ( ap @ V4abs2 @ ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ ( arr @ A_27c @ A_27d ) @ A_27c @ ( arr @ A_27a @ A_27b ) ) @ V1abs1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) ) @ V6f ) ) @ ( ap @ V2rep1 @ V7x ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EW__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 @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V7f2: $i] :
( ( mem @ V7f2 @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V8x1: $i] :
( ( mem @ V8x1 @ A_27a )
=> ! [V9x2: $i] :
( ( mem @ V9x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ V0R1 ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f1 ) @ V7f2 ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V8x1 ) @ V9x2 ) ) )
=> ( p @ ( ap @ ( ap @ V3R2 @ ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ V6f1 ) @ V8x1 ) ) @ ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ V7f2 ) @ V9x2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQ__IMPLIES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ( p @ V0P )
<=> ( p @ V1Q ) )
=> ( ( p @ V0P )
=> ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUALS__IMPLIES,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ A_27a )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ A_27a )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ A_27a )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ A_27a )
=> ( ( ( V0P = V2Q )
& ( V1P_27 = V3Q_27 ) )
=> ( ( V0P = V1P_27 )
=> ( V2Q = V3Q_27 ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ECONJ__IMPLIES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V0P )
=> ( p @ V2Q ) )
& ( ( p @ V1P_27 )
=> ( p @ V3Q_27 ) ) )
=> ( ( ( p @ V0P )
& ( p @ V1P_27 ) )
=> ( ( p @ V2Q )
& ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EDISJ__IMPLIES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V0P )
=> ( p @ V2Q ) )
& ( ( p @ V1P_27 )
=> ( p @ V3Q_27 ) ) )
=> ( ( ( p @ V0P )
| ( p @ V1P_27 ) )
=> ( ( p @ V2Q )
| ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EIMP__IMPLIES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1P_27: $i] :
( ( mem @ V1P_27 @ bool )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ bool )
=> ! [V3Q_27: $i] :
( ( mem @ V3Q_27 @ bool )
=> ( ( ( ( p @ V2Q )
=> ( p @ V0P ) )
& ( ( p @ V1P_27 )
=> ( p @ V3Q_27 ) ) )
=> ( ( ( p @ V0P )
=> ( p @ V1P_27 ) )
=> ( ( p @ V2Q )
=> ( p @ V3Q_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ENOT__IMPLIES,axiom,
! [V0P: $i] :
( ( mem @ V0P @ bool )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ bool )
=> ( ( ( p @ V1Q )
=> ( p @ V0P ) )
=> ( ~ ( p @ V0P )
=> ~ ( p @ V1Q ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUALS__EQUIV__IMPLIES,axiom,
! [A_27a: del,V0a1: $i] :
( ( mem @ V0a1 @ A_27a )
=> ! [V1a2: $i] :
( ( mem @ V1a2 @ A_27a )
=> ! [V2b1: $i] :
( ( mem @ V2b1 @ A_27a )
=> ! [V3b2: $i] :
( ( mem @ V3b2 @ A_27a )
=> ! [V4R: $i] :
( ( mem @ V4R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V4R ) )
=> ( ( ( p @ ( ap @ ( ap @ V4R @ V0a1 ) @ V1a2 ) )
& ( p @ ( ap @ ( ap @ V4R @ V2b1 ) @ V3b2 ) ) )
=> ( ( V0a1 = V2b1 )
=> ( p @ ( ap @ ( ap @ V4R @ V1a2 ) @ V3b2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EABSTRACT__RES__ABSTRACT,axiom,
! [A_27a: del,A_27b: del,A_27c: 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 ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V4f ) @ V5g ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V4f ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V5g ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__ABSTRACT__ABSTRACT,axiom,
! [A_27a: del,A_27b: del,A_27c: 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 ) ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5g: $i] :
( ( mem @ V5g @ ( arr @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V4f ) @ V5g ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V4f ) ) @ V5g ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__RES__ABSTRACT__LEFT,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 ) ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27a @ A_27b ) )
=> ! [V4x1: $i] :
( ( mem @ V4x1 @ A_27a )
=> ! [V5x2: $i] :
( ( mem @ V5x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f1 @ V4x1 ) ) @ ( ap @ V3f2 @ V5x2 ) ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V4x1 ) @ V4x1 ) ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V2f1 ) @ V4x1 ) ) @ ( ap @ V3f2 @ V5x2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__RES__ABSTRACT__RIGHT,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 ) ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ A_27a @ A_27b ) )
=> ! [V4x1: $i] :
( ( mem @ V4x1 @ A_27a )
=> ! [V5x2: $i] :
( ( mem @ V5x2 @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f1 @ V4x1 ) ) @ ( ap @ V3f2 @ V5x2 ) ) )
& ( p @ ( ap @ ( ap @ V0R1 @ V5x2 ) @ V5x2 ) ) )
=> ( p @ ( ap @ ( ap @ V1R2 @ ( ap @ V2f1 @ V4x1 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ERES__ABSTRACT @ A_27a @ A_27b ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R1 ) ) @ V3f2 ) @ V5x2 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__RES__FORALL,axiom,
! [A_27a: del,V0E: $i] :
( ( mem @ V0E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0E ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0E ) ) @ V1P ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__RES__EXISTS,axiom,
! [A_27a: del,V0E: $i] :
( ( mem @ V0E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0E ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0E ) ) @ V1P ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEQUIV__RES__EXISTS__UNIQUE,axiom,
! [A_27a: del,V0E: $i] :
( ( mem @ V0E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( c_2Equotient_2EEQUIV @ A_27a ) @ V0E ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0E ) ) @ V1P ) )
<=> ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V1P ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EFORALL__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V0P ) )
=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEXISTS__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V2x ) )
=> ( p @ ( ap @ V1Q @ V2x ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V0P ) )
=> ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V1Q ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__FORALL__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V2R @ V3x ) )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V2R ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V2R ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ A_27a @ bool ) )
=> ! [V2R: $i] :
( ( mem @ V2R @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V2R @ V3x ) )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V1Q @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V2R ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V2R ) @ V1Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELEFT__RES__FORALL__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1R @ V3x ) )
& ( ( p @ ( ap @ V2Q @ V3x ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1R ) @ V2Q ) )
=> ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERIGHT__RES__FORALL__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1R @ V3x ) )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A_27a ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__FORALL @ A_27a ) @ V1R ) @ V2Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ELEFT__RES__EXISTS__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1R @ V3x ) )
=> ( ( p @ ( ap @ V2Q @ V3x ) )
=> ( p @ ( ap @ V0P @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1R ) @ V2Q ) )
=> ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V0P ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERIGHT__RES__EXISTS__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V1R @ V3x ) )
& ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F @ A_27a ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS @ A_27a ) @ V1R ) @ V2Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2EEXISTS__UNIQUE__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1E: $i] :
( ( mem @ V1E @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1E ) @ V3x ) )
& ( p @ ( ap @ V2Q @ V3x ) ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1E ) @ V4x ) )
& ( p @ ( ap @ V2Q @ V4x ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1E ) @ V5y ) )
& ( p @ ( ap @ V2Q @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1E @ V4x ) @ V5y ) ) ) ) ) )
=> ( ( p @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V1E ) @ V2Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__UNIQUE__RESPECTS__REGULAR,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V1P ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__UNIQUE__REGULAR,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1R: $i] :
( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ V0P @ V3x ) )
=> ( p @ ( ap @ V2Q @ V3x ) ) ) )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5y: $i] :
( ( mem @ V5y @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1R ) @ V4x ) )
& ( p @ ( ap @ V2Q @ V4x ) )
& ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1R ) @ V5y ) )
& ( p @ ( ap @ V2Q @ V5y ) ) )
=> ( p @ ( ap @ ( ap @ V1R @ V4x ) @ V5y ) ) ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V1R ) ) @ V0P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V1R ) @ V2Q ) ) ) ) ) ) ) ).
thf(conj_thm_2Equotient_2ERES__EXISTS__UNIQUE__REGULAR__SAME,axiom,
! [A_27a: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ A_27a @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( 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 ) ) @ V1P ) @ V2Q ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2ERES__EXISTS__UNIQUE @ A_27a ) @ ( ap @ ( c_2Equotient_2Erespects @ A_27a @ bool ) @ V0R ) ) @ V1P ) )
=> ( p @ ( ap @ ( ap @ ( c_2Equotient_2ERES__EXISTS__EQUIV @ A_27a ) @ V0R ) @ V2Q ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------