ITP001 Axioms: ITP077^7.ax
%------------------------------------------------------------------------------
% File : ITP077^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : topology.ax [Gau19]
% : HL4077^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 110 ( 17 unt; 37 typ; 0 def)
% Number of atoms : 330 ( 39 equ; 4 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 1085 ( 4 ~; 1 |; 46 &; 942 @)
% ( 22 <=>; 70 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 9 avg; 942 nst)
% Number of types : 2 ( 1 usr)
% Number of type conns : 337 ( 337 >; 0 *; 0 +; 0 <<)
% Number of symbols : 38 ( 36 usr; 1 con; 0-5 aty)
% Number of variables : 318 ( 5 ^ 271 !; 9 ?; 318 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Etopology_2Etopology,type,
tyop_2Etopology_2Etopology: $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EBIGUNION,type,
c_2Epred__set_2EBIGUNION:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ECOMPL,type,
c_2Epred__set_2ECOMPL:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EEMPTY,type,
c_2Epred__set_2EEMPTY:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EGSPEC,type,
c_2Epred__set_2EGSPEC:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).
thf(c_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2EINSERT,type,
c_2Epred__set_2EINSERT:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EINTER,type,
c_2Epred__set_2EINTER:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epred__set_2EUNION,type,
c_2Epred__set_2EUNION:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Etopology_2Eclosed,type,
c_2Etopology_2Eclosed:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > $o ) ).
thf(c_2Etopology_2Eclosed__in,type,
c_2Etopology_2Eclosed__in:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Etopology_2Ehull,type,
c_2Etopology_2Ehull:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( A_27a > $o ) > A_27a > $o ) ).
thf(c_2Etopology_2Eistopology,type,
c_2Etopology_2Eistopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > $o ) ).
thf(c_2Etopology_2Elimpt,type,
c_2Etopology_2Elimpt:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Etopology_2Eneigh,type,
c_2Etopology_2Eneigh:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( tyop_2Epair_2Eprod @ ( A_27a > $o ) @ A_27a ) > $o ) ).
thf(c_2Etopology_2Eopen,type,
c_2Etopology_2Eopen:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > $o ) ).
thf(c_2Etopology_2Eopen__in,type,
c_2Etopology_2Eopen__in:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o ) ).
thf(c_2Etopology_2Etopology,type,
c_2Etopology_2Etopology:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > ( tyop_2Etopology_2Etopology @ A_27a ) ) ).
thf(c_2Etopology_2Etopspace,type,
c_2Etopology_2Etopspace:
!>[A_27a: $tType] : ( ( tyop_2Etopology_2Etopology @ A_27a ) > A_27a > $o ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Etopology_2Eistopology,axiom,
! [A_27a: $tType,V0L: ( A_27a > $o ) > $o] :
( ( c_2Etopology_2Eistopology @ A_27a @ V0L )
<=> ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) @ V0L )
& ! [V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V1s @ V0L )
& ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V0L ) )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) @ V0L ) )
& ! [V3k: ( A_27a > $o ) > $o] :
( ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V3k @ V0L )
=> ( c_2Ebool_2EIN @ ( A_27a > $o ) @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V3k ) @ V0L ) ) ) ) ).
thf(thm_2Etopology_2Etopology__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Etopology_2Etopology @ A_27a ) > ( A_27a > $o ) > $o] : ( c_2Ebool_2ETYPE__DEFINITION @ ( ( A_27a > $o ) > $o ) @ ( tyop_2Etopology_2Etopology @ A_27a ) @ ( c_2Etopology_2Eistopology @ A_27a ) @ V0rep ) ).
thf(thm_2Etopology_2Etopology__tybij,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Etopology @ A_27a @ ( c_2Etopology_2Eopen__in @ A_27a @ V0a ) )
= V0a )
& ! [V1r: ( A_27a > $o ) > $o] :
( ( c_2Etopology_2Eistopology @ A_27a @ V1r )
<=> ( ( c_2Etopology_2Eopen__in @ A_27a @ ( c_2Etopology_2Etopology @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Etopology_2Eopen__DEF,axiom,
! [A_27a: $tType,V0s: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen @ A_27a @ V0s )
= ( c_2Etopology_2Eopen__in @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Etopology_2Etopspace,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Etopspace @ A_27a @ V0top )
= ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V1s: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V1s @ ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s ) ) ) ) ) ).
thf(thm_2Etopology_2Eneigh,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1N: A_27a > $o,V2x: A_27a] :
( ( c_2Etopology_2Eneigh @ A_27a @ V0top @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ V1N @ V2x ) )
<=> ? [V3P: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V3P )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3P @ V1N )
& ( V3P @ V2x ) ) ) ).
thf(thm_2Etopology_2Eclosed__in,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) )
& ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) @ V1s ) ) ) ) ).
thf(thm_2Etopology_2Eclosed__DEF,axiom,
! [A_27a: $tType,V0s: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eclosed @ A_27a @ V0s )
= ( c_2Etopology_2Eclosed__in @ A_27a @ V0s @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Etopology_2Elimpt,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1x: A_27a,V2S_27: A_27a > $o] :
( ( c_2Etopology_2Elimpt @ A_27a @ V0top @ V1x @ V2S_27 )
<=> ! [V3N: A_27a > $o] :
( ( c_2Etopology_2Eneigh @ A_27a @ V0top @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ V3N @ V1x ) )
=> ? [V4y: A_27a] :
( ( (~) @ ( V1x = V4y ) )
& ( V2S_27 @ V4y )
& ( V3N @ V4y ) ) ) ) ).
thf(thm_2Etopology_2Ehull,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s )
= ( c_2Epred__set_2EBIGINTER @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2t: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V2t @ ( c_2Ebool_2E_2F_5C @ ( V0P @ V2t ) @ ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) ) ) ) ) ) ).
thf(thm_2Etopology_2EISTOPOLOGY__OPEN__IN,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] : ( c_2Etopology_2Eistopology @ A_27a @ ( c_2Etopology_2Eopen__in @ A_27a @ V0top ) ) ).
thf(thm_2Etopology_2ETOPOLOGY__EQ,axiom,
! [A_27a: $tType,V0top1: tyop_2Etopology_2Etopology @ A_27a,V1top2: tyop_2Etopology_2Etopology @ A_27a] :
( ( V0top1 = V1top2 )
<=> ! [V2s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top1 @ V2s )
= ( c_2Etopology_2Eopen__in @ A_27a @ V1top2 @ V2s ) ) ) ).
thf(thm_2Etopology_2Eopen__topspace,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen @ A_27a @ V0top )
=> ( ( c_2Etopology_2Etopspace @ A_27a @ V0top )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__CLAUSES,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EEMPTY @ A_27a ) )
& ! [V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) )
& ! [V3k: ( A_27a > $o ) > $o] :
( ! [V4s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4s @ V3k )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V4s ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V3k ) ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__SUBSET,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__EMPTY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] : ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Etopology_2EOPEN__IN__INTER,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__BIGUNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1k: ( A_27a > $o ) > $o] :
( ! [V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V1k )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2s ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1k ) ) ) ).
thf(thm_2Etopology_2EBIGUNION__2,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EBIGUNION @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V1t @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
= ( c_2Epred__set_2EUNION @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Etopology_2EOPEN__IN__UNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__TOPSPACE,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] : ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ).
thf(thm_2Etopology_2EOPEN__IN__BIGINTER,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ( (~)
@ ( V1s
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) )
& ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2t ) ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1s ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__SUBOPEN,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
<=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ? [V3t: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V3t )
& ( c_2Ebool_2EIN @ A_27a @ V2x @ V3t )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3t @ V1s ) ) ) ) ).
thf(thm_2Etopology_2EOPEN__OWN__NEIGH,axiom,
! [A_27a: $tType,V0S_27: A_27a > $o,V1top: tyop_2Etopology_2Etopology @ A_27a,V2x: A_27a] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V0S_27 )
& ( V0S_27 @ V2x ) )
=> ( c_2Etopology_2Eneigh @ A_27a @ V1top @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ V0S_27 @ V2x ) ) ) ).
thf(thm_2Etopology_2EOPEN__UNOPEN,axiom,
! [A_27a: $tType,V0S_27: A_27a > $o,V1top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V0S_27 )
<=> ( ( c_2Epred__set_2EBIGUNION @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( A_27a > $o ) @ ( A_27a > $o )
@ ^ [V2P: A_27a > $o] : ( c_2Epair_2E_2C @ ( A_27a > $o ) @ $o @ V2P @ ( c_2Ebool_2E_2F_5C @ ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V2P ) @ ( c_2Epred__set_2ESUBSET @ A_27a @ V2P @ V0S_27 ) ) ) ) )
= V0S_27 ) ) ).
thf(thm_2Etopology_2EOPEN__SUBOPEN,axiom,
! [A_27a: $tType,V0S_27: A_27a > $o,V1top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V0S_27 )
<=> ! [V2x: A_27a] :
( ( V0S_27 @ V2x )
=> ? [V3P: A_27a > $o] :
( ( V3P @ V2x )
& ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V3P )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3P @ V0S_27 ) ) ) ) ).
thf(thm_2Etopology_2EOPEN__NEIGH,axiom,
! [A_27a: $tType,V0S_27: A_27a > $o,V1top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V1top @ V0S_27 )
<=> ! [V2x: A_27a] :
( ( V0S_27 @ V2x )
=> ? [V3N: A_27a > $o] :
( ( c_2Etopology_2Eneigh @ A_27a @ V1top @ ( c_2Epair_2E_2C @ ( A_27a > $o ) @ A_27a @ V3N @ V2x ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V3N @ V0S_27 ) ) ) ) ).
thf(thm_2Etopology_2Eclosed__topspace,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eclosed @ A_27a @ V0top )
=> ( ( c_2Etopology_2Etopspace @ A_27a @ V0top )
= ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__OPEN__IN__COMPL,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eclosed @ A_27a @ V0top )
=> ! [V1s: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
= ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2ECOMPL @ A_27a @ V1s ) ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__SUBSET,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__EMPTY,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] : ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__TOPSPACE,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] : ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__UNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__BIGINTER,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1k: ( A_27a > $o ) > $o] :
( ( ( (~)
@ ( V1k
= ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) )
& ! [V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2s @ V1k )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V2s ) ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V1k ) ) ) ).
thf(thm_2Etopology_2EBIGINTER__2,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1t: A_27a > $o] :
( ( c_2Epred__set_2EBIGINTER @ A_27a @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V0s @ ( c_2Epred__set_2EINSERT @ ( A_27a > $o ) @ V1t @ ( c_2Epred__set_2EEMPTY @ ( A_27a > $o ) ) ) ) )
= ( c_2Epred__set_2EINTER @ A_27a @ V0s @ V1t ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__INTER,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EINTER @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__CLOSED__IN__EQ,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
<=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) )
& ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) @ V1s ) ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__CLOSED__IN,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) )
=> ( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
= ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2Etopology_2Etopspace @ A_27a @ V0top ) @ V1s ) ) ) ) ).
thf(thm_2Etopology_2EOPEN__IN__DIFF,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__DIFF,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1s )
& ( c_2Etopology_2Eopen__in @ A_27a @ V0top @ V2t ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EDIFF @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2ECLOSED__IN__BIGUNION,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a,V1s: ( A_27a > $o ) > $o] :
( ( ( c_2Epred__set_2EFINITE @ ( A_27a > $o ) @ V1s )
& ! [V2t: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V2t @ V1s )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V2t ) ) )
=> ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ ( c_2Epred__set_2EBIGUNION @ A_27a @ V1s ) ) ) ).
thf(thm_2Etopology_2ECLOSED__LIMPT,axiom,
! [A_27a: $tType,V0top: tyop_2Etopology_2Etopology @ A_27a] :
( ( c_2Etopology_2Eclosed @ A_27a @ V0top )
=> ! [V1S_27: A_27a > $o] :
( ( c_2Etopology_2Eclosed__in @ A_27a @ V0top @ V1S_27 )
<=> ! [V2x: A_27a] :
( ( c_2Etopology_2Elimpt @ A_27a @ V0top @ V2x @ V1S_27 )
=> ( V1S_27 @ V2x ) ) ) ) ).
thf(thm_2Etopology_2EHULL__P,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ( V0P @ V1s )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s )
= V1s ) ) ).
thf(thm_2Etopology_2EP__HULL,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ! [V2f: ( A_27a > $o ) > $o] :
( ! [V3s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V2f )
=> ( V0P @ V3s ) )
=> ( V0P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V2f ) ) )
=> ( V0P @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) ) ) ).
thf(thm_2Etopology_2EHULL__EQ,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ! [V2f: ( A_27a > $o ) > $o] :
( ! [V3s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V2f )
=> ( V0P @ V3s ) )
=> ( V0P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V2f ) ) )
=> ( ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s )
= V1s )
<=> ( V0P @ V1s ) ) ) ).
thf(thm_2Etopology_2EHULL__HULL,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) ) ).
thf(thm_2Etopology_2EHULL__SUBSET,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) ) ).
thf(thm_2Etopology_2EHULL__MONO,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) ) ) ).
thf(thm_2Etopology_2EHULL__ANTIMONO,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1Q: ( A_27a > $o ) > $o,V2s: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ ( A_27a > $o ) @ V0P @ V1Q )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V1Q @ V2s ) @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) ) ) ).
thf(thm_2Etopology_2EHULL__MINIMAL,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
& ( V0P @ V2t ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ V2t ) ) ).
thf(thm_2Etopology_2ESUBSET__HULL,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( V0P @ V2t )
=> ( ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ V2t )
= ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2EHULL__UNIQUE,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V2t )
& ( V0P @ V2t )
& ! [V3t_27: A_27a > $o] :
( ( ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ V3t_27 )
& ( V0P @ V3t_27 ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ V3t_27 ) ) )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s )
= V2t ) ) ).
thf(thm_2Etopology_2EHULL__UNION__SUBSET,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] : ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) ) @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) ) ) ).
thf(thm_2Etopology_2EHULL__UNION,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) ) ) ) ).
thf(thm_2Etopology_2EHULL__UNION__LEFT,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) @ V2t ) ) ) ).
thf(thm_2Etopology_2EHULL__UNION__RIGHT,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ V2t ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EUNION @ A_27a @ V1s @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) ) ) ) ).
thf(thm_2Etopology_2EHULL__REDUNDANT__EQ,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1a: A_27a,V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V1a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) )
<=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) ) ) ).
thf(thm_2Etopology_2EHULL__REDUNDANT,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1a: A_27a,V2s: A_27a > $o] :
( ( c_2Ebool_2EIN @ A_27a @ V1a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EINSERT @ A_27a @ V1a @ V2s ) )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) ) ) ).
thf(thm_2Etopology_2EHULL__INDUCT,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1p: A_27a > $o,V2s: A_27a > $o] :
( ( ! [V3x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V3x @ V2s )
=> ( V1p @ V3x ) )
& ( V0P
@ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
@ ^ [V4x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V4x @ ( V1p @ V4x ) ) ) ) )
=> ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) )
=> ( V1p @ V5x ) ) ) ).
thf(thm_2Etopology_2EHULL__INC,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ V1s )
=> ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) ) ) ).
thf(thm_2Etopology_2EHULL__IMAGE__SUBSET,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1f: A_27a > A_27a,V2s: A_27a > $o] :
( ( ( V0P @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) )
& ! [V3s: A_27a > $o] :
( ( V0P @ V3s )
=> ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V3s ) ) ) )
=> ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V2s ) ) @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) ) ) ) ).
thf(thm_2Etopology_2EHULL__IMAGE__GALOIS,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1f: A_27a > A_27a,V2g: A_27a > A_27a,V3s: A_27a > $o] :
( ( ! [V4s: A_27a > $o] : ( V0P @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V4s ) )
& ! [V5s: A_27a > $o] :
( ( V0P @ V5s )
=> ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V5s ) ) )
& ! [V6s: A_27a > $o] :
( ( V0P @ V6s )
=> ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V2g @ V6s ) ) )
& ! [V7s: A_27a > $o,V8t: A_27a > $o] :
( ( c_2Epred__set_2ESUBSET @ A_27a @ V7s @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V2g @ V8t ) )
= ( c_2Epred__set_2ESUBSET @ A_27a @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V7s ) @ V8t ) ) )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V3s ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V3s ) ) ) ) ).
thf(thm_2Etopology_2EHULL__IMAGE,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1f: A_27a > A_27a,V2s: A_27a > $o] :
( ( ! [V3s: A_27a > $o] : ( V0P @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V3s ) )
& ! [V4s: A_27a > $o] :
( ( V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V4s ) )
= ( V0P @ V4s ) )
& ! [V5x: A_27a,V6y: A_27a] :
( ( ( V1f @ V5x )
= ( V1f @ V6y ) )
=> ( V5x = V6y ) )
& ! [V7y: A_27a] :
? [V8x: A_27a] :
( ( V1f @ V8x )
= V7y ) )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ V2s ) )
= ( c_2Epred__set_2EIMAGE @ A_27a @ A_27a @ V1f @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2s ) ) ) ) ).
thf(thm_2Etopology_2EIS__HULL,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o] :
( ! [V2f: ( A_27a > $o ) > $o] :
( ! [V3s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V3s @ V2f )
=> ( V0P @ V3s ) )
=> ( V0P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V2f ) ) )
=> ( ( V0P @ V1s )
<=> ? [V4t: A_27a > $o] :
( V1s
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ V4t ) ) ) ) ).
thf(thm_2Etopology_2EHULLS__EQ,axiom,
! [A_27a: $tType,V0P: ( A_27a > $o ) > $o,V1s: A_27a > $o,V2t: A_27a > $o] :
( ( ! [V3f: ( A_27a > $o ) > $o] :
( ! [V4s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4s @ V3f )
=> ( V0P @ V4s ) )
=> ( V0P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V3f ) ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V1s @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) )
& ( c_2Epred__set_2ESUBSET @ A_27a @ V2t @ ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s ) ) )
=> ( ( c_2Etopology_2Ehull @ A_27a @ V0P @ V1s )
= ( c_2Etopology_2Ehull @ A_27a @ V0P @ V2t ) ) ) ).
thf(thm_2Etopology_2EHULL__P__AND__Q,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1P: ( A_27a > $o ) > $o,V2Q: ( A_27a > $o ) > $o] :
( ( ! [V3f: ( A_27a > $o ) > $o] :
( ! [V4s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V4s @ V3f )
=> ( V1P @ V4s ) )
=> ( V1P @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V3f ) ) )
& ! [V5f: ( A_27a > $o ) > $o] :
( ! [V6s: A_27a > $o] :
( ( c_2Ebool_2EIN @ ( A_27a > $o ) @ V6s @ V5f )
=> ( V2Q @ V6s ) )
=> ( V2Q @ ( c_2Epred__set_2EBIGINTER @ A_27a @ V5f ) ) )
& ! [V7s: A_27a > $o] :
( ( V2Q @ V7s )
=> ( V2Q @ ( c_2Etopology_2Ehull @ A_27a @ V1P @ V7s ) ) ) )
=> ( ( c_2Etopology_2Ehull @ A_27a
@ ^ [V8x: A_27a > $o] : ( c_2Ebool_2E_2F_5C @ ( V1P @ V8x ) @ ( V2Q @ V8x ) )
@ V0s )
= ( c_2Etopology_2Ehull @ A_27a @ V1P @ ( c_2Etopology_2Ehull @ A_27a @ V2Q @ V0s ) ) ) ) ).
%------------------------------------------------------------------------------