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 : 82 ( 0 unt; 0 def)
% Number of atoms : 502 ( 35 equ)
% Maximal formula atoms : 18 ( 6 avg)
% Number of connectives : 423 ( 3 ~; 0 |; 53 &)
% ( 23 <=>; 344 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 9 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 37 ( 37 usr; 2 con; 0-3 aty)
% Number of variables : 293 ( 285 !; 8 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(ne_ty_2Etopology_2Etopology,axiom,
! [A0] :
( ne(A0)
=> ne(ty_2Etopology_2Etopology(A0)) ) ).
fof(mem_c_2Etopology_2Eclosed,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eclosed(A_27a),arr(ty_2Etopology_2Etopology(A_27a),bool)) ) ).
fof(mem_c_2Etopology_2Eclosed__in,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eclosed__in(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Etopology_2Ehull,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Ehull(A_27a),arr(arr(arr(A_27a,bool),bool),arr(arr(A_27a,bool),arr(A_27a,bool)))) ) ).
fof(mem_c_2Etopology_2Eistopology,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eistopology(A_27a),arr(arr(arr(A_27a,bool),bool),bool)) ) ).
fof(mem_c_2Etopology_2Elimpt,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Elimpt(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(A_27a,arr(arr(A_27a,bool),bool)))) ) ).
fof(mem_c_2Etopology_2Eneigh,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eneigh(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(ty_2Epair_2Eprod(arr(A_27a,bool),A_27a),bool))) ) ).
fof(mem_c_2Etopology_2Eopen,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eopen(A_27a),arr(ty_2Etopology_2Etopology(A_27a),bool)) ) ).
fof(mem_c_2Etopology_2Eopen__in,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Eopen__in(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(arr(A_27a,bool),bool))) ) ).
fof(mem_c_2Etopology_2Etopology,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Etopology(A_27a),arr(arr(arr(A_27a,bool),bool),ty_2Etopology_2Etopology(A_27a))) ) ).
fof(mem_c_2Etopology_2Etopspace,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Etopology_2Etopspace(A_27a),arr(ty_2Etopology_2Etopology(A_27a),arr(A_27a,bool))) ) ).
fof(ax_thm_2Etopology_2Eistopology,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0L] :
( 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] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(ax_thm_2Etopology_2Etopology__TY__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ? [V0rep] :
( 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)) ) ) ).
fof(ax_thm_2Etopology_2Etopology__tybij,axiom,
! [A_27a] :
( ne(A_27a)
=> ( ! [V0a] :
( mem(V0a,ty_2Etopology_2Etopology(A_27a))
=> ap(c_2Etopology_2Etopology(A_27a),ap(c_2Etopology_2Eopen__in(A_27a),V0a)) = V0a )
& ! [V1r] :
( 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 ) ) ) ) ).
fof(conj_thm_2Etopology_2EISTOPOLOGY__OPEN__IN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(c_2Etopology_2Eistopology(A_27a),ap(c_2Etopology_2Eopen__in(A_27a),V0top))) ) ) ).
fof(conj_thm_2Etopology_2ETOPOLOGY__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top1] :
( mem(V0top1,ty_2Etopology_2Etopology(A_27a))
=> ! [V1top2] :
( mem(V1top2,ty_2Etopology_2Etopology(A_27a))
=> ( V0top1 = V1top2
<=> ! [V2s] :
( 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)) ) ) ) ) ) ) ).
fof(ax_thm_2Etopology_2Eopen__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(lameq_f2014,axiom,
! [A_27a,V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] : ap(f2014(A_27a,V0top),V1s) = ap(ap(c_2Epair_2E_2C(arr(A_27a,bool),bool),V1s),ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s)) ) ).
fof(ax_thm_2Etopology_2Etopspace,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( 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)),f2014(A_27a,V0top))) ) ) ).
fof(conj_thm_2Etopology_2Eopen__topspace,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( 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) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__CLAUSES,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a)))
& ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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] :
( mem(V3k,arr(arr(A_27a,bool),bool))
=> ( ! [V4s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1k] :
( mem(V1k,arr(arr(A_27a,bool),bool))
=> ( ! [V2s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EBIGUNION__2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__TOPSPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__SUBOPEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V0top),V1s))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(c_2Ebool_2EIN(A_27a),V2x),V1s))
=> ? [V3t] :
( 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)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Etopology_2Eneigh,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1N] :
( mem(V1N,arr(A_27a,bool))
=> ! [V2x] :
( 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] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__OWN__NEIGH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0S_27] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ! [V2x] :
( 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))) ) ) ) ) ) ).
fof(lameq_f2015,axiom,
! [A_27a,V0S_27] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ! [V2P] : ap(f2015(A_27a,V0S_27,V1top),V2P) = 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))) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__UNOPEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0S_27] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top] :
( 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)),f2015(A_27a,V0S_27,V1top))) = V0S_27 ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__SUBOPEN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0S_27] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0S_27,V2x))
=> ? [V3P] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__NEIGH,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0S_27] :
( mem(V0S_27,arr(A_27a,bool))
=> ! [V1top] :
( mem(V1top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(ap(c_2Etopology_2Eopen__in(A_27a),V1top),V0S_27))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(V0S_27,V2x))
=> ? [V3N] :
( 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)) ) ) ) ) ) ) ) ).
fof(ax_thm_2Etopology_2Eclosed__in,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(ax_thm_2Etopology_2Eclosed__DEF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( 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))) ) ) ) ).
fof(conj_thm_2Etopology_2Eclosed__topspace,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( 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) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__OPEN__IN__COMPL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0top))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__EMPTY,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),c_2Epred__set_2EEMPTY(A_27a))) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__TOPSPACE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),ap(c_2Etopology_2Etopspace(A_27a),V0top))) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__BIGINTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1k] :
( mem(V1k,arr(arr(A_27a,bool),bool))
=> ( ( V1k != c_2Epred__set_2EEMPTY(arr(A_27a,bool))
& ! [V2s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EBIGINTER__2,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1t] :
( 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) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__CLOSED__IN,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EOPEN__IN__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__DIFF,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__IN__BIGUNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1s] :
( mem(V1s,arr(arr(A_27a,bool),bool))
=> ( ( p(ap(c_2Epred__set_2EFINITE(arr(A_27a,bool)),V1s))
& ! [V2t] :
( 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))) ) ) ) ) ).
fof(ax_thm_2Etopology_2Elimpt,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ! [V1x] :
( mem(V1x,A_27a)
=> ! [V2S_27] :
( mem(V2S_27,arr(A_27a,bool))
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),V0top),V1x),V2S_27))
<=> ! [V3N] :
( 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] :
( mem(V4y,A_27a)
& V1x != V4y
& p(ap(V2S_27,V4y))
& p(ap(V3N,V4y)) ) ) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ECLOSED__LIMPT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0top] :
( mem(V0top,ty_2Etopology_2Etopology(A_27a))
=> ( p(ap(c_2Etopology_2Eclosed(A_27a),V0top))
=> ! [V1S_27] :
( mem(V1S_27,arr(A_27a,bool))
=> ( p(ap(ap(c_2Etopology_2Eclosed__in(A_27a),V0top),V1S_27))
<=> ! [V2x] :
( mem(V2x,A_27a)
=> ( p(ap(ap(ap(c_2Etopology_2Elimpt(A_27a),V0top),V2x),V1S_27))
=> p(ap(V1S_27,V2x)) ) ) ) ) ) ) ) ).
fof(lameq_f2016,axiom,
! [A_27a,V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V2t] : ap(f2016(A_27a,V1s,V0P),V2t) = 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))) ) ) ).
fof(ax_thm_2Etopology_2Ehull,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( 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)),f2016(A_27a,V1s,V0P))) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__P,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( p(ap(V0P,V1s))
=> ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V1s) = V1s ) ) ) ) ).
fof(conj_thm_2Etopology_2EP__HULL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__HULL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( 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) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( 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))) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__MONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__ANTIMONO,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1Q] :
( mem(V1Q,arr(arr(A_27a,bool),bool))
=> ! [V2s] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__MINIMAL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2ESUBSET__HULL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__UNIQUE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( p(ap(ap(c_2Epred__set_2ESUBSET(A_27a),V1s),V2t))
& p(ap(V0P,V2t))
& ! [V3t_27] :
( 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 ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__UNION__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__UNION,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__UNION__LEFT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__UNION__RIGHT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( 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))) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__REDUNDANT__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2s] :
( 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) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__REDUNDANT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1a] :
( mem(V1a,A_27a)
=> ! [V2s] :
( 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) ) ) ) ) ) ).
fof(lameq_f2017,axiom,
! [A_27a,V1p] :
( mem(V1p,arr(A_27a,bool))
=> ! [V4x] : ap(f2017(A_27a,V1p),V4x) = ap(ap(c_2Epair_2E_2C(A_27a,bool),V4x),ap(V1p,V4x)) ) ).
fof(conj_thm_2Etopology_2EHULL__INDUCT,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1p] :
( mem(V1p,arr(A_27a,bool))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3x] :
( 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),f2017(A_27a,V1p)))) )
=> ! [V5x] :
( 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)) ) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__INC,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2x] :
( 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))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__IMAGE__SUBSET,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V2s)))
& ! [V3s] :
( 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)))) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__IMAGE__GALOIS,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2g] :
( mem(V2g,arr(A_27a,A_27a))
=> ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> ( ( ! [V4s] :
( mem(V4s,arr(A_27a,bool))
=> p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V4s))) )
& ! [V5s] :
( 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] :
( 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] :
( mem(V7s,arr(A_27a,bool))
=> ! [V8t] :
( 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)) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULL__IMAGE,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1f] :
( mem(V1f,arr(A_27a,A_27a))
=> ! [V2s] :
( mem(V2s,arr(A_27a,bool))
=> ( ( ! [V3s] :
( mem(V3s,arr(A_27a,bool))
=> p(ap(V0P,ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V3s))) )
& ! [V4s] :
( 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] :
( mem(V5x,A_27a)
=> ! [V6y] :
( mem(V6y,A_27a)
=> ( ap(V1f,V5x) = ap(V1f,V6y)
=> V5x = V6y ) ) )
& ! [V7y] :
( mem(V7y,A_27a)
=> ? [V8x] :
( 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)) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EIS__HULL,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ( ! [V2f] :
( mem(V2f,arr(arr(A_27a,bool),bool))
=> ( ! [V3s] :
( 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] :
( mem(V4t,arr(A_27a,bool))
& V1s = ap(ap(c_2Etopology_2Ehull(A_27a),V0P),V4t) ) ) ) ) ) ) ).
fof(conj_thm_2Etopology_2EHULLS__EQ,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(arr(A_27a,bool),bool))
=> ! [V1s] :
( mem(V1s,arr(A_27a,bool))
=> ! [V2t] :
( mem(V2t,arr(A_27a,bool))
=> ( ( ! [V3f] :
( mem(V3f,arr(arr(A_27a,bool),bool))
=> ( ! [V4s] :
( 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) ) ) ) ) ) ).
fof(lameq_f2018,axiom,
! [A_27a,V1P] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ! [V2Q] :
( mem(V2Q,arr(arr(A_27a,bool),bool))
=> ! [V8x] : ap(f2018(A_27a,V1P,V2Q),V8x) = ap(ap(c_2Ebool_2E_2F_5C,ap(V1P,V8x)),ap(V2Q,V8x)) ) ) ).
fof(conj_thm_2Etopology_2EHULL__P__AND__Q,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0s] :
( mem(V0s,arr(A_27a,bool))
=> ! [V1P] :
( mem(V1P,arr(arr(A_27a,bool),bool))
=> ! [V2Q] :
( mem(V2Q,arr(arr(A_27a,bool),bool))
=> ( ( ! [V3f] :
( mem(V3f,arr(arr(A_27a,bool),bool))
=> ( ! [V4s] :
( 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] :
( mem(V5f,arr(arr(A_27a,bool),bool))
=> ( ! [V6s] :
( 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] :
( 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),f2018(A_27a,V1P,V2Q)),V0s) = ap(ap(c_2Etopology_2Ehull(A_27a),V1P),ap(ap(c_2Etopology_2Ehull(A_27a),V2Q),V0s)) ) ) ) ) ) ).
%------------------------------------------------------------------------------