ITP001 Axioms: ITP077^5.ax
%------------------------------------------------------------------------------
% File : ITP077^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : topology^2.ax [Gau20]
% : HL4077^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 87 ( 0 unt; 11 typ; 0 def)
% Number of atoms : 1628 ( 30 equ; 0 cnn)
% Maximal formula atoms : 63 ( 18 avg)
% Number of connectives : 2945 ( 3 ~; 0 |; 53 &;2607 @)
% ( 23 <=>; 259 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 15 avg;2607 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 11 ( 11 >; 0 *; 0 +; 0 <<)
% Number of symbols : 37 ( 36 usr; 25 con; 0-2 aty)
% Number of variables : 279 ( 5 ^ 266 !; 8 ?; 279 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Etopology_2Etopology,type,
ty_2Etopology_2Etopology: del > del ).
thf(tp_c_2Etopology_2Eclosed,type,
c_2Etopology_2Eclosed: del > $i ).
thf(mem_c_2Etopology_2Eclosed,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eclosed @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ bool ) ) ).
thf(tp_c_2Etopology_2Eclosed__in,type,
c_2Etopology_2Eclosed__in: del > $i ).
thf(mem_c_2Etopology_2Eclosed__in,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Etopology_2Ehull,type,
c_2Etopology_2Ehull: del > $i ).
thf(mem_c_2Etopology_2Ehull,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Ehull @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2Etopology_2Eistopology,type,
c_2Etopology_2Eistopology: del > $i ).
thf(mem_c_2Etopology_2Eistopology,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eistopology @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ bool ) ) ).
thf(tp_c_2Etopology_2Elimpt,type,
c_2Etopology_2Elimpt: del > $i ).
thf(mem_c_2Etopology_2Elimpt,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Elimpt @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ) ).
thf(tp_c_2Etopology_2Eneigh,type,
c_2Etopology_2Eneigh: del > $i ).
thf(mem_c_2Etopology_2Eneigh,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eneigh @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( ty_2Epair_2Eprod @ ( arr @ A_27a @ bool ) @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Etopology_2Eopen,type,
c_2Etopology_2Eopen: del > $i ).
thf(mem_c_2Etopology_2Eopen,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eopen @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ bool ) ) ).
thf(tp_c_2Etopology_2Eopen__in,type,
c_2Etopology_2Eopen__in: del > $i ).
thf(mem_c_2Etopology_2Eopen__in,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ) ).
thf(tp_c_2Etopology_2Etopology,type,
c_2Etopology_2Etopology: del > $i ).
thf(mem_c_2Etopology_2Etopology,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( ty_2Etopology_2Etopology @ A_27a ) ) ) ).
thf(tp_c_2Etopology_2Etopspace,type,
c_2Etopology_2Etopspace: del > $i ).
thf(mem_c_2Etopology_2Etopspace,axiom,
! [A_27a: del] : ( mem @ ( c_2Etopology_2Etopspace @ A_27a ) @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(ax_thm_2Etopology_2Eistopology,axiom,
! [A_27a: del,V0L: $i] :
( ( mem @ V0L @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eistopology @ A_27a ) @ V0L ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) @ V0L ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V1s ) @ V0L ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V0L ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) @ V0L ) ) ) ) )
& ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V3k ) @ V0L ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V3k ) ) @ V0L ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Etopology__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Etopology_2Etopology @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) @ ( ty_2Etopology_2Etopology @ A_27a ) ) @ ( c_2Etopology_2Eistopology @ A_27a ) ) @ V0rep ) ) ) ).
thf(ax_thm_2Etopology_2Etopology__tybij,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( ap @ ( c_2Etopology_2Etopology @ A_27a ) @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eistopology @ A_27a ) @ V1r ) )
<=> ( ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopology @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(conj_thm_2Etopology_2EISTOPOLOGY__OPEN__IN,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( p @ ( ap @ ( c_2Etopology_2Eistopology @ A_27a ) @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) ) ) ) ).
thf(conj_thm_2Etopology_2ETOPOLOGY__EQ,axiom,
! [A_27a: del,V0top1: $i] :
( ( mem @ V0top1 @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1top2: $i] :
( ( mem @ V1top2 @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( V0top1 = V1top2 )
<=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top1 ) @ V2s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top2 ) @ V2s ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Eopen__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eopen @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Etopspace,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top )
= ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V1s: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V1s ) @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2Eopen__topspace,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eopen @ A_27a ) @ V0top ) )
=> ( ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__CLAUSES,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) )
& ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) )
& ! [V3k: $i] :
( ( mem @ V3k @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4s ) @ V3k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V4s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V3k ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__INTER,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__BIGUNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V1k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2s ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EBIGUNION__2,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V1t ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V0s ) @ V1t ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__UNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__TOPSPACE,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__BIGINTER,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1s ) )
& ( V1s
!= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2t ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__SUBOPEN,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ? [V3t: $i] :
( ( mem @ V3t @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V3t ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3t ) @ V1s ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Eneigh,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1N: $i] :
( ( mem @ V1N @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ V1N ) @ V2x ) ) )
<=> ? [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V3P ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3P ) @ V1N ) )
& ( p @ ( ap @ V3P @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__OWN__NEIGH,axiom,
! [A_27a: del,V0S_27: $i] :
( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
=> ! [V1top: $i] :
( ( mem @ V1top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V0S_27 ) )
& ( p @ ( ap @ V0S_27 @ V2x ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ V1top ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ V0S_27 ) @ V2x ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__UNOPEN,axiom,
! [A_27a: del,V0S_27: $i] :
( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
=> ! [V1top: $i] :
( ( mem @ V1top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V0S_27 ) )
<=> ( ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2P: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V2P ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V2P ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2P ) @ V0S_27 ) ) ) ) ) )
= V0S_27 ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__SUBOPEN,axiom,
! [A_27a: del,V0S_27: $i] :
( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
=> ! [V1top: $i] :
( ( mem @ V1top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V0S_27 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0S_27 @ V2x ) )
=> ? [V3P: $i] :
( ( mem @ V3P @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ V3P @ V2x ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V3P ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3P ) @ V0S_27 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__NEIGH,axiom,
! [A_27a: del,V0S_27: $i] :
( ( mem @ V0S_27 @ ( arr @ A_27a @ bool ) )
=> ! [V1top: $i] :
( ( mem @ V1top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V1top ) @ V0S_27 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ V0S_27 @ V2x ) )
=> ? [V3N: $i] :
( ( mem @ V3N @ ( arr @ A_27a @ bool ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ V1top ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ V3N ) @ V2x ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V3N ) @ V0S_27 ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Eclosed__in,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) @ V1s ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Eclosed__DEF,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eclosed @ A_27a ) @ V0s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0s ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Etopology_2Eclosed__topspace,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eclosed @ A_27a ) @ V0top ) )
=> ( ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__OPEN__IN__COMPL,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eclosed @ A_27a ) @ V0top ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2ECOMPL @ A_27a ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__SUBSET,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__EMPTY,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__TOPSPACE,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__UNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__BIGINTER,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1k: $i] :
( ( mem @ V1k @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( V1k
!= ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) )
& ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2s ) @ V1k ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V2s ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V1k ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EBIGINTER__2,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1t: $i] :
( ( mem @ V1t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V0s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ ( arr @ A_27a @ bool ) ) @ V1t ) @ ( c_2Epred__set_2EEMPTY @ ( arr @ A_27a @ bool ) ) ) ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V0s ) @ V1t ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__INTER,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINTER @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN__EQ,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
<=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ ( ap @ ( c_2Etopology_2Etopspace @ A_27a ) @ V0top ) ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EOPEN__IN__DIFF,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__DIFF,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1s ) )
& ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eopen__in @ A_27a ) @ V0top ) @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epred__set_2EDIFF @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__IN__BIGUNION,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( arr @ A_27a @ bool ) ) @ V1s ) )
& ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V2t ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V2t ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ ( ap @ ( c_2Epred__set_2EBIGUNION @ A_27a ) @ V1s ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Elimpt,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2S_27: $i] :
( ( mem @ V2S_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ A_27a ) @ V0top ) @ V1x ) @ V2S_27 ) )
<=> ! [V3N: $i] :
( ( mem @ V3N @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eneigh @ A_27a ) @ V0top ) @ ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ A_27a ) @ V3N ) @ V1x ) ) )
=> ? [V4y: $i] :
( ( mem @ V4y @ A_27a )
& ( V1x != V4y )
& ( p @ ( ap @ V2S_27 @ V4y ) )
& ( p @ ( ap @ V3N @ V4y ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ECLOSED__LIMPT,axiom,
! [A_27a: del,V0top: $i] :
( ( mem @ V0top @ ( ty_2Etopology_2Etopology @ A_27a ) )
=> ( ( p @ ( ap @ ( c_2Etopology_2Eclosed @ A_27a ) @ V0top ) )
=> ! [V1S_27: $i] :
( ( mem @ V1S_27 @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Etopology_2Eclosed__in @ A_27a ) @ V0top ) @ V1S_27 ) )
<=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Etopology_2Elimpt @ A_27a ) @ V0top ) @ V2x ) @ V1S_27 ) )
=> ( p @ ( ap @ V1S_27 @ V2x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Etopology_2Ehull,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s )
= ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a )
@ ( ap @ ( c_2Epred__set_2EGSPEC @ ( arr @ A_27a @ bool ) @ ( arr @ A_27a @ bool ) )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V2t: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ ( arr @ A_27a @ bool ) @ bool ) @ V2t ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V0P @ V2t ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__P,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ V1s ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s )
= V1s ) ) ) ) ).
thf(conj_thm_2Etopology_2EP__HULL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ V2f ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V2f ) ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__EQ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ V2f ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V2f ) ) ) ) )
=> ( ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s )
= V1s )
<=> ( p @ ( ap @ V0P @ V1s ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__HULL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__MONO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__ANTIMONO,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1Q: $i] :
( ( mem @ V1Q @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ ( arr @ A_27a @ bool ) ) @ V0P ) @ V1Q ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V1Q ) @ V2s ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__MINIMAL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ V0P @ V2t ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2ESUBSET__HULL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ V2t ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ V2t ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__UNIQUE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V2t ) )
& ( p @ ( ap @ V0P @ V2t ) )
& ! [V3t_27: $i] :
( ( mem @ V3t_27 @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ V3t_27 ) )
& ( p @ ( ap @ V0P @ V3t_27 ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ V3t_27 ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s )
= V2t ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__UNION__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__UNION,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__UNION__LEFT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__UNION__RIGHT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ V2t ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EUNION @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__REDUNDANT__EQ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) )
<=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__REDUNDANT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1a ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__INDUCT,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1p: $i] :
( ( mem @ V1p @ ( arr @ A_27a @ bool ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3x: $i] :
( ( mem @ V3x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ V2s ) )
=> ( p @ ( ap @ V1p @ V3x ) ) ) )
& ( p
@ ( ap @ V0P
@ ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V4x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V4x ) @ ( ap @ V1p @ V4x ) ) ) ) ) ) )
=> ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V5x ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) )
=> ( p @ ( ap @ V1p @ V5x ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__INC,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2x: $i] :
( ( mem @ V2x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ V1s ) )
=> ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__IMAGE__SUBSET,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) )
& ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ V3s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V3s ) ) ) ) ) )
=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V2s ) ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__IMAGE__GALOIS,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2g: $i] :
( ( mem @ V2g @ ( arr @ A_27a @ A_27a ) )
=> ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V4s ) ) ) )
& ! [V5s: $i] :
( ( mem @ V5s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ V5s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V5s ) ) ) ) )
& ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ V6s ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V2g ) @ V6s ) ) ) ) )
& ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27a @ bool ) )
=> ! [V8t: $i] :
( ( mem @ V8t @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V7s ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V2g ) @ V8t ) ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V7s ) ) @ V8t ) ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V3s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V3s ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__IMAGE,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ! [V2s: $i] :
( ( mem @ V2s @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V3s ) ) ) )
& ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V4s ) ) )
<=> ( p @ ( ap @ V0P @ V4s ) ) ) )
& ! [V5x: $i] :
( ( mem @ V5x @ A_27a )
=> ! [V6y: $i] :
( ( mem @ V6y @ A_27a )
=> ( ( ( ap @ V1f @ V5x )
= ( ap @ V1f @ V6y ) )
=> ( V5x = V6y ) ) ) )
& ! [V7y: $i] :
( ( mem @ V7y @ A_27a )
=> ? [V8x: $i] :
( ( mem @ V8x @ A_27a )
& ( ( ap @ V1f @ V8x )
= V7y ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ V2s ) )
= ( ap @ ( ap @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a ) @ V1f ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2s ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EIS__HULL,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ( ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V3s: $i] :
( ( mem @ V3s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V3s ) @ V2f ) )
=> ( p @ ( ap @ V0P @ V3s ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V2f ) ) ) ) )
=> ( ( p @ ( ap @ V0P @ V1s ) )
<=> ? [V4t: $i] :
( ( mem @ V4t @ ( arr @ A_27a @ bool ) )
& ( V1s
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V4t ) ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULLS__EQ,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V1s: $i] :
( ( mem @ V1s @ ( arr @ A_27a @ bool ) )
=> ! [V2t: $i] :
( ( mem @ V2t @ ( arr @ A_27a @ bool ) )
=> ( ( ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4s ) @ V3f ) )
=> ( p @ ( ap @ V0P @ V4s ) ) ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V3f ) ) ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V1s ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Epred__set_2ESUBSET @ A_27a ) @ V2t ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V1s )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V0P ) @ V2t ) ) ) ) ) ) ).
thf(conj_thm_2Etopology_2EHULL__P__AND__Q,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1P: $i] :
( ( mem @ V1P @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ! [V2Q: $i] :
( ( mem @ V2Q @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ( ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V4s: $i] :
( ( mem @ V4s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V4s ) @ V3f ) )
=> ( p @ ( ap @ V1P @ V4s ) ) ) )
=> ( p @ ( ap @ V1P @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V3f ) ) ) ) )
& ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) )
=> ( ! [V6s: $i] :
( ( mem @ V6s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ ( arr @ A_27a @ bool ) ) @ V6s ) @ V5f ) )
=> ( p @ ( ap @ V2Q @ V6s ) ) ) )
=> ( p @ ( ap @ V2Q @ ( ap @ ( c_2Epred__set_2EBIGINTER @ A_27a ) @ V5f ) ) ) ) )
& ! [V7s: $i] :
( ( mem @ V7s @ ( arr @ A_27a @ bool ) )
=> ( ( p @ ( ap @ V2Q @ V7s ) )
=> ( p @ ( ap @ V2Q @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V1P ) @ V7s ) ) ) ) ) )
=> ( ( ap
@ ( ap @ ( c_2Etopology_2Ehull @ A_27a )
@ ( lam @ ( arr @ A_27a @ bool )
@ ^ [V8x: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ V1P @ V8x ) ) @ ( ap @ V2Q @ V8x ) ) ) )
@ V0s )
= ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V1P ) @ ( ap @ ( ap @ ( c_2Etopology_2Ehull @ A_27a ) @ V2Q ) @ V0s ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------